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#72: Completion Nonstandard Analysis on 07.03.2018

Preliminary remarks: In the following, we shall use the definitions established in the chapters on Set Theory and Topology, and we will usually take \(m, n \in {}^{\omega}\mathbb{N}^{*}\). We seek to study integration and differentiation on an arbitrary non-empty subset \(A\) of the sets \({}^{(\omega)}\mathbb{C}^{n}\) or \({}^{(\omega)}\mathbb{R}^{n}\) for arbitrary \(n\), which we shall combine with the notation \({}^{(\omega)}\mathbb{K}^{n} = {}^{(\omega)}\mathbb{K}\times ... \times{}^{(\omega)}\mathbb{K}\), where each \(\mathbb{K}\) represents either \(\mathbb{C}\) or \(\mathbb{R}\) in some arbitrary order. In particular, we will consider inconcrete and infinite sets that are conventionally non-measurable, and functions that are conventionally discontinuous. Every element not in the image set is replaced by the neighbouring element in the target set, and if multiple choices are possible one single choice is selected. If we do not do this, the mapping is not (meaningfully) defined. In the following, \(||\cdot||\) denotes the Euclidean norm. This can easily be generalised to other sets and norms if the latter are equivalent.

Definition: We describe neighbouring points in \(A\) by means of the irreflexive symmetric neighbourhood relation \(B \subseteq {A}^{2}\). The function \(\gamma: C \rightarrow A \subseteq \mathbb{C}{}^{n}\), where \(C \subseteq \mathbb{R}\) is \(h\)-homogeneous and h is infinitesimal, is called a path if \(||\gamma(x) - \gamma(y)||\) is infinitesimal for all neighbouring points \(x, y \in C\) and (\(\gamma(x), \gamma(y)) \in B\). The neighbourhood relations of \(B\) in \(A\) are systematically written as (predecessor, successor) with the notation \(({z}_{0}, \curvearrowright {z}_{0})\) or \((\curvearrowleft {z}_{0}, {z}_{0})\), where \(\curvearrowright\) is pronounced "post" and \(\curvearrowleft\) is pronounced "pre". This applies analogously to the neighbourhood relation \(D \subseteq C{}^{2}\).

Definition: Let \({z}_{0} \in A \subseteq \mathbb{K}^{n}\) and \(f: A \rightarrow {}^{(c)}\mathbb{K}^{m}\). In the following, we will omit proofs for predecessors, since they are analogous to the proofs for successors. We say that \(f\) is \(\alpha B\)-successor-continuous in \({z}_{0}\) in the direction \(\curvearrowright B {z}_{0}\) if for infinitesimal \(\alpha \in {}^{(\omega)}\mathbb{R}{}_{>0}\), we have that:\[||f(\curvearrowright B {z}_{0}) - f({z}_{0})|| < \alpha\]If the exact modulus of \(\alpha\) does not matter, \(\alpha\) may be omitted in the notation. If \(f\) is \(\alpha B\)-successor-continuous for all \({z}_{0}\) and \(\curvearrowright B {z}_{0}\), we simply say that it is \(\alpha B\)-continuous. We say that \(\alpha\) is the degree of continuity. If the inequality only holds for \(\alpha = 1/c\), we simply say that \(f\) is (\(B\)-successor-)continuous. The property of \(\alpha B\)-predecessor-continuity is defined analogously.

Remark: In practice, we should choose \(\alpha\) by estimating \(f\) (for example after considering any jump discontinuities). If \(B\) is obvious or irrelevant, it may be omitted. Below, we systematically omit it when \(B = {}^{(\omega)}\mathbb{K}{}^{2n}\).

Example: The function \(f: \mathbb{R} \rightarrow \{\pm 1\}\) with \(f(x) = {(-1)}^{x/d0}\) is nowhere successor-continuous on \(\mathbb{R}\), but its absolute value is (cf. Number Theory). Here, \(x/d0\) is an integer since \(\mathbb{R}\) is \(d0\)-homogeneous. If we set \(f(x) = 1\) for rational \(x\) and \(= -1\) otherwise, then \(f(x)\) is partially \(d0\)-successor-continuous on non-rational numbers, unlike the conventional notion of continuity.

Definition: For \(f: A \rightarrow {}^{(\omega)}\mathbb{K}{}^{m}\), we define\[{d}_{\curvearrowright B z}f(z) := f(\curvearrowright B z) - f(z)\]to be the \(B\)-successor-differential of \(f\) in the direction \(\curvearrowright B z\) for \(z \in A\). If dim \(A = n\), then we can specify \({d}_{\curvearrowright B z}f(z)\) as \(d((\curvearrowright B){z}_{1}, ... , (\curvearrowright B){z}_{n})f(z\)). If \(f\) is the identity, i.e. \(f(z) = z\), then we can write \({d}_{\curvearrowright B z}Bz\) instead of \({d}_{\curvearrowright B z}f(z)\). If \(A\) or \(\curvearrowright B z\) is obvious or irrelevant, it can be omitted. The conventional real case can be defined analogously to the above.

Remark: If the modulus of the \(B\)-successor-differential of \(f\) in the direction \(\curvearrowright B z\) at \(z \in A\) is smaller than \(\alpha\) and infinitesimal, then \(f\) is also \(\alpha B\)-successor-continuous at that point.

Definition: An (infinitely) real-valued function with arguments \(\in {}^{(\omega)}\mathbb{K}{}^{n}\) is said to be convex (concave) if the line segment between any two points on the graph of the function lies above (below) or on the graph. It is said to be strictly convex (concave) if "or on" can be omitted.

Definition: The \(m\) arithmetic means of all \({f}_{k}(\curvearrowright B z)\) of \(f(z)\) give the \(m\) averaged normed tangential normal vectors of \(m\) (uniquely determined) hyperplanes, defining the \(mn\) continuous partial derivatives of the Jacobian matrix of \(f\), which is not necessarily continuous. The hyperplanes are taken to pass through \({f}_{k}(\curvearrowright B z)\) and \(f(z)\) translated towards 0. We can minimise the moduli of their coefficients by solving in a very simple linear programme (cf. Linear Programming).

From this we obtain directly the

Theorem: Every in \(A \subseteq {}^{(\omega)}\mathbb{K}{}^{n}\) convex resp. concave function \(f: A \rightarrow {}^{(\omega)}\mathbb{R}\) is \(\alpha B\)-successor-continuous and \(B\)-successor-differentiable.\(\square\)

Example of a Peano curve (from [739], p. 188): "Consider the even, periodic function \(g: \mathbb{R} \rightarrow \mathbb{R}\) with period 2 and image [0, 1] defined by\[{g}(t)=\left\{ \begin{matrix} 0 \\ 3t-1 \\ 1 \\ \end{matrix}\begin{matrix} for\,0\le t<\tfrac{1}{3} \\ for\,\tfrac{1}{3}\le t<\tfrac{2}{3} \\ for\,\tfrac{2}{3}\le t\le 1. \\ \end{matrix} \right.\,\]It is clear that g is fully specified by this definition, and continuous. Now let the function \(\Phi: I = [0, 1] \rightarrow \mathbb{R}^{2}\) be defined by\[\phi(t) = \left( {\sum\limits_{k = 0}^{\infty} {\frac{{g({4^{2k}}t)}}{{{2^{k + 1}}}},} \sum\limits_{k = 0}^{\infty} {\frac{{g({4^{2k + 1}}t)}}{{{2^{k + 1}}}}} } \right)."\]The function \(\Phi\) is at least continuous since the sums are ultimately locally linear functions in \(t\), when \(\infty\) is replaced by \(\acute{\omega}\). It would however be an error to believe that [0, 1] can be bijectively mapped onto \({[0, 1]}^{2}\), in this way, e.g. because the powers of four in \(g\), and the values 0 and 1 taken by \(g\) in two sub-intervals thin out \({[0, 1]}^{2}\) so much that a bijection is clearly impossible. Restricting the proof to rational points only is simply insufficient.

Definition: A point \(x (\curvearrowright x)\) of a function \(f: A \subseteq {}^{\omega}\mathbb{R} \rightarrow {}^{\omega}\mathbb{R}\) is said to be a right (left) jump discontinuity of \(s := |f(\curvearrowright x) - f(x)|\) upwards (downwards), or vice versa if \(s > 1/|{}^{\omega}\mathbb{N}^{*}|\).

Theorem: A monotone function \(f: [a, b] \rightarrow {}^{\omega}\mathbb{R}\) has at most \(|{}^{\omega}\mathbb{N}|| {}^{\omega}\mathbb{Z}^{*}| - 1\) jump discontinuities.

Proof: Between \(-|{}^{\omega}\mathbb{N}^{*}|\) and \(|{}^{\omega}\mathbb{N}^{*}|\), at most \(|{}^{\omega}\mathbb{N}^{*}|| {}^{\omega}\mathbb{Z}^{*}|\) jump discontinuities with a jump of \(1/|{}^{\omega}\mathbb{N}^{*}|\) are possible, totalling a maximum of \(|{}^{\omega}\mathbb{Z}^{*}| - 1\) when all viewed together. If the function does not decrease at non-discontinuities, like a step function, then the claim follows after correctly accounting for the ends of the number line \({}^{\omega}\mathbb{R}\).\(\square\)

Remark: This theorem corrects Froda's theorem and makes it more precise. If we precede each set by the superscript \({}^{c}\), we obtain the statement for conventional sets.

Definition: The partial derivative in the direction \(\curvearrowright B {z}_{k}\) of \(F: A \rightarrow {}^{(\omega)}\mathbb{K}\) at \(z = ({z}_{1}, ..., {z}_{n}) \in A \subseteq {}^{(\omega)}\mathbb{K}^{n}\) with \(k \in [1, n]\mathbb{N}\) is defined as\[\frac{\partial B\,F(z)}{\partial B\,{{z}_{k}}}:=\frac{F({{z}_{1}},\,...,\,\curvearrowright B\,{{z}_{k}},\,...,\,{{z}_{n}})-F(z)}{\curvearrowright B\,{{z}_{k}}-{{z}_{k}}}.\]With this notation, if the function \(f\) satisfies \(f = ({f}_{1}, ..., {f}_{n}): A \rightarrow {}^{(\omega)}\mathbb{K}^{n}\) with \(z \in A \subseteq {}^{(\omega)}\mathbb{K}^{n}\)\[f(z)=\left( \frac{F(\curvearrowright B{{z}_{1}},{{z}_{2}},...,{{z}_{n}})-F({{z}_{1}},...,{{z}_{n}})}{(\curvearrowright B{{z}_{1}}-{{z}_{1}})},...,\frac{F({{z}_{1}},...,{{z}_{n-1}},\curvearrowright B{{z}_{n}})-F({{z}_{1}},...,{{z}_{n}})}{(\curvearrowright B{{z}_{n}}-{{z}_{n}})} \right)=\left( \frac{\partial B{{F}_{1}}(z)}{\partial B{{z}_{1}}},\,\,...\,\,,\,\,\frac{\partial B{{F}_{n}}(z)}{\partial B{{z}_{n}}} \right)=grad{{B}_{\curvearrowright Bz}}\,F(z)\,=\,\nabla {{B}_{\curvearrowright Bz}}\,F(z),\]then \(f(z)\) is said to be exact \(B\)-successor-derivative \({F'}_{\curvearrowright B z} B(z)\) or the exact \(B\)-successor-gradient \({grad}_{\curvearrowright B z} F(z)\) in the direction \(\curvearrowright B z\) on \(A\) of the function \(F\) at \(z\), which is said to be exactly \(B\)-differentiable at \(z\) in the direction \(\curvearrowright B z\), provided that each quotient exists in \({}^{(\omega)}\mathbb{K}\). \(\nabla\) is the Nabla operator. If this definition is satisfied for every \(z \in A\), then \(F\) is said to be an exactly \(B\)-differentiable \(B\)-antiderivative of \(f\). On the (conventionally) (infinite) reals, the left and right \(B\)-antiderivatives \({F}_{l}(x)\) and \({F}_{r}(x)\) at \(x \in {}^{(\omega)}\mathbb{R}\) distinguish between the cases of the corresponding \(B\)-derivatives.

If \(A\) or \(\curvearrowright B z\) are obvious from context or irrelevant, they can be omitted. The conventional case may be obtained analogously to the above. In the case \(n = 1\), we say that \({F'}_{r}B(x)\) is the right exact \(B\)-derivative for \(\curvearrowright B x > x \in {}^{(\omega)}\mathbb{R}\) and that \({F'}_{l}B(x)\) is the left exact \(B\)-derivative for \(\curvearrowright B x < x\). If both directions have the same value, we say that \(F'B(z)\) is the exact derivative (when \(A ={}^{c}\mathbb{C}\) and \(n = 1\), this reduces to the conventional case where \(F\) is holomorphic).

Remark: Clearly, the \(B\)-antiderivatives of a given function only differ by a (conventional) (infinite) complex or (infinite) real additive constant. The \(B\)-antiderivatives of discontinuous functions can typically only be derived by adding and appropriately recombining easier \(\alpha B\)-continuous functions (e.g. by reversing the rules of differentiation).

Chain rule: For \(x \in A \subseteq {}^{(\omega)}\mathbb{R}, B \subseteq {A}^{2}, f: A \rightarrow C \subseteq {}^{(\omega)}\mathbb{R}, D \subseteq {C}^{2}, g: C \rightarrow {}^{(\omega)}\mathbb{R}\), choosing \(f(\curvearrowright B x) = \curvearrowright D f(x)\)), we have that:\[{g'}_{r}B(f(x)) = {g'}_{r}D(f(x)) {f'}_{r}B(x).\]Proof:\[{{{g}'}_{r}}B(f(x))=\frac{g(f(\curvearrowright Bx))-g(f(x))}{f(\curvearrowright Bx)-f(x)}\frac{f(\curvearrowright Bx)-f(x)}{\curvearrowright Bx-x}=\frac{g(\curvearrowright Df(x))-g(f(x))}{\curvearrowright Df(x)-f(x)}{{{f}'}_{r}}B(x)={{{g}'}_{r}}D(f(x)){{{f}'}_{r}}B(x).\square\]Remark: The chain rule and the rules stated below can be extended to (infinite) complex sets and left exact derivatives. We will subsequently omit the sets and neighbourhood relations. Let \(f\) and \(g\) be right exactly differentiable (infinite) real functions at \(x \in A \subseteq {}^{(\omega)}\mathbb{R}\).

Product rule:\((fg){'}_{r}(x) = {f'}_{r}(x) g(x) + f(\curvearrowright x) {g'}_{r}(x)= {f'}_{r}(x) g(\curvearrowright x) + f(x) {g'}_{r}(x).\)

Proof: Add and subtract \(f(\curvearrowright x) g(x)\) resp. \(f(x) g(\curvearrowright x)\) in the numerator.\(\square\)

Quotient rule: Suppose that the denominators of the following quotients are non-zero. Then:\[\left( \frac{f}{g} \right)_{r}^{\prime }(x)=\frac{{{{{f}'}}_{r}}(x)\,g(x)-f(x)\,{{{{g}'}}_{r}}(x)}{g(x)\,g(\curvearrowright x)}=\frac{{{{{f}'}}_{r}}(x)\,g(\curvearrowright x)-f(\curvearrowright x)\,{{{{g}'}}_{r}}(x)}{g(x)\,g(\curvearrowright x)}.\square\]Proof: Add and subtract \(f(x) g(x)\) resp. \(f(\curvearrowright x) g(\curvearrowright x)\) in the numerator.\(\square\)

Remark: For the product and quotient rule to be as precise as the conventional one, the arguments and function values must belong to a smaller level of infinity than \(1/d0\), and \(f\) and \(g\) must be sufficiently (\(\alpha\)-) continuous at \(x \in A\) (i.e. \(\alpha\) must be sufficiently small) to allow \(\curvearrowright x\) to be replaced by \(x\). An analogous principle holds for infinitesimal arguments.

Remark: The right exact derivative of the inverse function\[{f}^{-1}{'}_{r}(y) = 1/{f'}_{r}(x)\]can be derived from \(y = f(x)\) and the identity \(x = {f}^{-1}(f(x))\) using the chain rule with an equal level of precision. L'Hôpital's rule also makes sense for (\(\alpha\)-) continuous functions \(f\) and \(g\) such that \(f(v) = g(v) = 0\) with \(v \in A\) and \(g(\curvearrowright v) \ne 0\), and may be stated as:\[\frac{f(\curvearrowright v)}{g(\curvearrowright v)}=\frac{f(\curvearrowright v)-f(v)}{g(\curvearrowright v)-g(v)}=\frac{{{{{f}'}}_{r}}(v)}{{{{{g}'}}_{r}}(v)}.\]

Remark: Differentiability is thus easy to establish. In the (conventional) (infinite) real case, we can set\[{{{F}'}_{b}}B(v)\,:=\,\frac{F(\curvearrowright B\,v)-F(\curvearrowleft B\,v)}{\curvearrowright B\,v-\curvearrowleft B\,v}\] wherever this quotient is defined. This is especially useful when \(\curvearrowright B v - v = v - \curvearrowleft B v\), and the combined derivatives both have the same sign. This definition has the advantage of allowing us to view \({F'}_{b} \; B(v)\) as the "tangent slope" at the point \(v\), especially when \(F\) is \(\alpha B\)-continuous at \(v\). It also results in simpler rules of differentiation, in particular since a derivative value of 0 is most suitable for cases with opposite signs (see below). In other cases, we simply calculate the arithmetic mean of both exact derivatives. This can be extended to the (conventional) complex numbers analogously.

Definition: Given \(z \in A \subseteq {}^{(\omega)}\mathbb{K}^{n}\), we define \[\int\limits_{z\in A}{f(z)dBz:=\sum\limits_{z\in A}{f(z)(\curvearrowright B\,z-z)}}\]to be the exact \(B\)-integral of the vector field \(f = ({f}_{1}, ..., {f}_{n}): A \rightarrow {}^{(\omega)}\mathbb{K}^{n}\) on \(A\) and we say that \(f(z)\) is \(B\)-integrable. If this requires us to remove at least one point from \(A\), then we say that the exact \(B\)-integral is improper.
For \(\gamma: [a, b[C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}^{n}, C \subseteq \mathbb{R}\), and \(f = ({f}_{1}, ..., {f}_{n}): A \rightarrow {}^{(\omega)}\mathbb{K}^{n}\), we say that\[\int\limits_{\gamma }{f(\zeta )dB\zeta =}\int\limits_{t\in [a,b[C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}\]where \(dDt > 0, \curvearrowright D t \in ]a, b]C\), choosing \(\curvearrowright B \gamma(t) = \gamma(\curvearrowright D t)\), since \(\zeta = \gamma(t)\) and \(dB\zeta = \gamma(\curvearrowright D t) - \gamma(t) = {\gamma'}_{\curvearrowright }D(t) dDt\) (i.e. in particular for \(C = \mathbb{R}, B\) maximal in \(\mathbb{C}^{2}\), and \(D\) maximal in \(\mathbb{R}^{2})\), is the exact \(B\)-line integral of the vector field \(f\) along the path \(\gamma\). Improper exact \(B\)-line integrals are defined analogously to exact \(B\)-integrals, except that only interval end points may be removed from \([a, b[C\).

Remark: The value of the exact line integral on \({}^{(c)}\mathbb{K}\) is usually consistent with the conventional line integral; however, \(f\) does not need to be continuous and the proper \(B\)-line integral exists always. It can easily be seen that the exact \(B\)-line integral is linear and monotone in the (conventional) (infinite) real case. The art of integration lies in correctly combining the summands of the defining sum.

Definition: For all \(x \in V\) of an \(h\)-homogeneous \(n\)-volume \(V \subseteq [{a}_{1}, {b}_{1}] \times...\times [{a}_{n}, {b}_{n}] \subseteq {}^{(\omega)}\mathbb{R}^{n}\) with \(B = {B}_{1}\times...\times{B}_{n}, {B}_{k} \subseteq {[{a}_{k}, {b}_{k}]}^{2}\) and \(|{dB}_{k}{x}_{k}| = h\) for all \(k \in [1, n]\mathbb{N}\), we say that\[\int\limits_{x\in V}{f(x){dBx}}:=\int\limits_{x\in V}{f(x)dB({{x}_{1}},\,...,{{x}_{n}})}:=\int\limits_{{{a}_{n}}}^{{{b}_{n}}}{...\int\limits_{{{a}_{1}}}^{{{b}_{1}}}{f(x)d{{B}_{1}}{{x}_{1}}\,...\,d{{B}_{n}}{{x}_{n}}}}\]is the exact \(B\)-volume integral of the \(B\)-volume integrable function \(f: {}^{(\omega)}\mathbb{R}^{n} \rightarrow {}^{(\omega)}\mathbb{R}\) with \(f(x) := 0\) for all \(x \in {}^{(\omega)}\mathbb{R}^{n} \setminus V\). Improper exact \(B\)-volume integrals are defined analogously to exact \(B\)-integrals.

Remark: Because \(\mathbb{C}\) and \(\mathbb{R}^{2}\) are isomorphic, we have something similar in the complex case and\[\int\limits_{x\in V}{dBx={{\mu }_{h}}(V)}.\]Remark: An alternative definition of the exact volume integral may be given as described further below. However, the original definitions are easier to manipulate. In some cases, suitable Landau notation may be useful. If the result of differentiation lies outside of the domain, it should be replaced by the closest number within the domain. If this is not uniquely determined, the result can either be given as the set of all such numbers, or we can select the preferred result (e.g. according to a uniform rule).

Example: Let \([a, b[h{}^{\omega}\mathbb{Z}\) be a non-empty \(h\)-homogeneous subset of \([a, b[{}^{\omega}\mathbb{R}\), and write \(B \subseteq [a, b[h{}^{\omega}\mathbb{Z} \times ]a, b]h{}^{\omega}\mathbb{Z}\). Now let \({T}_{r}\) be a right \(B\)-antiderivative of a not necessarily convergent Taylor series \(t\) on \([a, b[h{}^{\omega}\mathbb{Z}\) and define \(f(x) := t(x) + {\varepsilon(-1)}^{x/h}\) for conventionally real \(x\) and \(\varepsilon \ge 1/c\). For \(h = 1/c\), \(f\) is nowhere continuous, and thus is conventionally nowhere differentiable or integrable on \([a, b[h{}^{\omega}\mathbb{Z}\), but for all \(h\) we have the following exact result\[f_{r}^{\prime }B(x)=t_{r}^{\prime }B(x)-\frac{2\varepsilon {{(-1)}^{\tfrac{x}{h}}}}{dBx}\]and\[\int\limits_{x \in [a,b[h{}^{\omega }\mathbb{Z}}f(x)dBx={{T}_{r}}(b)-{{T}_{r}}(a)+\frac{\varepsilon }{2}\left( {{(-1)}^{\tfrac{a}{h}}}-{{(-1)}^{\tfrac{b}{h}}} \right).\]Example: The interval \(Q := [0, 1[\mathbb{Q}\) has measure \({\mu}_{d0}(Q) = \frac{1}{2}\) (cf. Set Theory). Consider the function \(q: [0, 1[\rightarrow \{0, 1\}\) defined by \(q(x) = 1\) for \(x \in Q\) and \(q(x) = 0\) for \(x \in [0, 1[ \; \setminus \; Q\). Then\[\int\limits_{x\in Q}{q(x)dx=\sum\limits_{x=0}^{1}{q(x)dx}={\frac{1}{2}}}.\]Example: The middle-thirds Cantor set \({C}_{\frac{1}{3}}\) has measure \({\mu}_{d0}({C}_{\frac{1}{3}}) = {(\frac{2}{3})}^{\acute{\omega}}\). Consider the function\(c: [0, 1] \rightarrow \{0, {(\frac{2}{3})}^{-\acute{\omega}}\}\) defined by \(c(x) = {(\frac{2}{3})}^{-\acute{\omega}}\) for \(x \in {C}_{\frac{1}{3}}\) and \(c(x) = 0\) for \(x \in [0, 1] \setminus {C}_{\frac{1}{3}}\). Then\[\int\limits_{x \in {{C}_{\frac{1}{3}}}}{c(x)dx=\sum\limits_{x=0}^{1}{c(x)dx}}={{\left(\frac{2}{3}\right)}^{-\acute{\omega}}}{{\mu }_{d0}}\left( {{C}_{}} \right)=1.\]Remark: The sets \(Q\) and \({C}_{\frac{1}{3}}\) are conventionally not measurable. Thus, exact integration is more general than Riemann, Lebesgue(-Stieltjes) integration and other types of integral, which only exist on conventionally measurable sets, and function values may skip values, which are not measured then. We only consider simple examples here for purposes of illustration, and more complex examples can of course be found.

Definition: A sequence \(({a}_{k})\) with members \({a}_{k}\) is a mapping from \({}^{(\omega)}\mathbb{Z}\) to \({}^{(\omega)}\mathbb{C}^{m}: k \mapsto {a}_{k}\). A series is a equence \(({s}_{k})\) with \(m \in {}^{(\omega)}\mathbb{Z}\) and partial sums\[{{s}_{k}}=\sum\limits_{j=m}^{k}{{{a}_{j}}}.\]

Remark: Thus, in particular, the Riemann series theorem is false, since when summing positive summands to a desired value we are forced to add negative values until the original sum is attained, and vice versa. The same is true in the case that we obtain a smaller or larger value than the sum of the positive or negative terms, since the remainders almost fully cancel, and so on. We must avoid choosing arbitrary methods to deal with infinities if we wish to avoid making mistakes. We may not neglect the existence of something because it is stored at infinity.

Remark: The associative, commutative, and distributive laws imply that sums can be arbitrarily rearranged. If we take care to calculate it correctly (using Landau symbols), the B-volume integral satisfies the following theorem:

Fubini's theorem: For \(X, Y \subseteq {}^{(\omega)}\mathbb{K}\), \(f: X\times Y \rightarrow {}^{(\omega)}\mathbb{K}\) satisfies\[\int\limits_{Y}{\int\limits_{X}{f(x,\,y)dBx\,}dBy}=\int\limits_{X\times Y}{f(x,\,y)dB(x,\,y)}=\int\limits_{X}{\int\limits_{Y}{f(x,\,y)dBy\,}dBx}.\]Proof: Reorder the sums corresponding to these integrals.\(\square\)

Example: Since\[\int\limits_{[a,\,b[\times [r,\,s[}{\frac{\left( {{x}^{2}}-{{y}^{2}} \right)}{{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}}{{d}^{2}}(x,\,y)}=\int\limits_{a}^{b}{\left. \frac{ydx}{{{x}^{2}}+{{y}^{2}}} \right|_{r}^{s}}=-\int\limits_{r}^{s}{\left. \frac{xdy}{{{x}^{2}}+{{y}^{2}}} \right|_{a}^{b}}=\arctan \frac{s}{b}-\arctan \frac{r}{b}+\arctan \frac{s}{a}-\arctan \frac{r}{a}\]by the principle of latest substitution (see below), we obtain the (improper) integral\[I(a,b):=\int\limits_{[a,\,b{{[}^{2}}}{\frac{\left( {{x}^{2}}-{{y}^{2}} \right)}{{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}}{{d}^{2}}(x,\,y)}=\arctan \frac{b}{b}-\arctan \frac{a}{b}+\arctan \frac{b}{a}-\arctan \frac{a}{a}=\frac{\pi }{2}-\frac{\pi }{2}=0\]and not\[I(0,1)=\int\limits_{0}^{1}{\int\limits_{0}^{1}{\frac{\left( {{x}^{2}}-{{y}^{2}} \right)}{{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}}dy\,dx}}=\int\limits_{0}^{1}{\frac{dx}{1+{{x}^{2}}}}=\frac{\pi }{4}\ne -\frac{\pi }{4}=-\int\limits_{0}^{1}{\frac{dy}{1+{{y}^{2}}}}=\int\limits_{0}^{1}{\int\limits_{0}^{1}{\frac{\left( {{x}^{2}}-{{y}^{2}} \right)}{{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}}dx\,dy}}=I(0,1).\]Finiteness criterion for series: The Euclidean norm of the partial sum with the largest index max \(k\) of a series \(({s}_{k})\), where \(k\) and \(j\) are natural numbers, is finite if and only if\[\left\| {{s}_{\max k}} \right\|=\sum\limits_{j=0}^{\max j}{{{\left( -1 \right)}^{j}}{{d}_{j}}}\]holds for a monotonically decreasing sequence \(({d}_{j}) \in {}^{c}\mathbb{R}_{\ge 0}\).

Proof: The sum is bounded below by 0 and bounded above by \({d}_{0}\). The claim follows directly from the ability to arbitrarily rearrange summands, sort them according to their signs and sizes, and recombine them or split them into separate sums.\(\square\)

Example: From the alternating harmonic series, it follows that

\[\sum\limits_{k=1}^{2\left\lfloor \acute{\omega}/2 \right\rfloor }{{{\left( -1 \right)}^{k}}}\left(\acute{\omega}-\tfrac{1}{k} \right)+\frac{1-{{\left( -1 \right)}^{\acute{\omega}}}}{2}=\ln 2.\]Definition: The following rearrangement for \({a}_{j}, {b}_{k} \in {}^{(\omega)}\mathbb{K}\) gives the series product and corrects the Cauchy product:\[\sum\limits_{j=1}^{\acute{\omega}}{{{a}_{j}}}\sum\limits_{k=1}^{\acute{\omega}}{{{b}_{k}}}=\sum\limits_{j=1}^{\acute{\omega}}{\left( \sum\limits_{k=1}^{j}{\left( {{a}_{k}}{{b}_{j-k+1}}+{{a}_{\acute{\omega}-k+1}}{{b}_{\acute{\omega}-j+k}} \right)}-{{a}_{j}}{{b}_{\acute{\omega}-j+1}} \right)}.\]Example: For following series product (cf. [763], p. 61 f.), we have that:\[{{\left( \sum\limits_{j=1}^{\acute{\omega}}{\tfrac{{{(-1)}^{j}}}{\sqrt{j}}} \right)}^{2}}=\sum\limits_{j=1}^{\acute{\omega}}{\left( \tfrac{{{(-1)}^{\acute{\omega}}}}{\sqrt{j(\acute{\omega}-j+1)}}-\sum\limits_{k=1}^{j}{\left( \tfrac{{{(-1)}^{j}}}{\sqrt{k(j-k+1)}}+\tfrac{{{(-1)}^{j}}}{\sqrt{(\acute{\omega}-k+1)(\acute{\omega}-j+k)}} \right)} \right)}={{\left( \left( \sqrt{2}-1 \right)\zeta \left( \tfrac{1}{2} \right) \right)}^{2}}.\]Example: The signum function \(sgn\) yields the following series product (cf. [763], p. 62):\[\sum\limits_{j=0}^{\acute{\omega}}{{{2}^{{{j}^{sgn(j)}}}}}\sum\limits_{k=0}^{\acute{\omega}}{sgn(k-\gamma )}=(\acute{\omega}-1){{2}^{\acute{\omega}+1}}\gg -2.\]Definition: A sequence \(({a}_{k})\) with \(k \in {}^{(\omega)}\mathbb{N}^{*}, {a}_{k} \in {}^{(\omega)}\mathbb{C}\) and \(\alpha \in ]0, 1/c]\) is said to be \(\alpha\)-convergent to \(a \in {}^{(\omega)}\mathbb{C}\) if there exists \(m \in {}^{(\omega)}\mathbb{N}^{*}\) satisfying \(|{a}_{k} - a| < \alpha\) for all \({a}_{k}\) with \(k \ge m\) such that the difference max \(k - m\) is not too large. The set \(\alpha\)-\(A\) of all such \(a\) is called set of \(\alpha\)-limit values of \(({a}_{k})\). An appropriately and uniquely determined representative of this set (e.g. the final value or mean value) is called the \(\alpha\)-limit value \(\alpha\)-\(a\). In the special case \(a = 0\), we say that the sequence is a zero sequence. If the inequality only holds for \(\alpha = 1/c\), the \(\alpha\)- is omitted from the notation.

Remark: We will usually choose \(k\) to be maximal and \(\alpha\) to be minimal. Conventional limit values are often only precise to less than \(\mathcal{O}(1/\acute{\omega})\) uand in general are too imprecise, since they are often e.g. (arbitrarily) algebraic (of a certain degree) or transcendental. The conventional formulation of the definition of conventional convergence, which always requires infinitely many or almost all members of the sequence to have an arbitrarily small distance from the limit value and only allows finitely many to have larger distance, needs to be extended, since otherwise only the largest index of each sequence is taken into account and considered to be relevant (cf. [813], p. 144). Only then is the monotone convergence valid (cf. [813], p. 155).

Remark: The statement that each positive number may be represented by a fully determined, unique, infinite decimal fraction is baseless because of the fundamental theorem of set theory (see p. 27 f.). Furthermore, any proof claiming that, for \(\varepsilon \in {}^{(\omega)}\mathbb{R}_{>0}\) - in particular whenever the phrase "for all conventionally reals \(\varepsilon > 0\)" is used - there exists a real number \(\varepsilon/r\) with real \(r \in {}^{(\omega)}\mathbb{R}_{>1}\), is false, because we can simply set \(\varepsilon := \; \curvearrowright 0\), or become stuck in an infinite regression. Therefore, in the \(\varepsilon\delta\)-definition of the limit value (it is questionable that \(\delta\) exists, p. 235 f.) and in the \(\varepsilon\delta\)-definition of continuity (see p. 215) (consider for example the real function that doubles every real value but is not even uniformly continuous), \(\varepsilon\) must be restricted to specific multiples of \(\curvearrowright 0\).

Remark: We need not to consider uniform continuity, since in general we can set \(\delta := \; \curvearrowright 0\) and \(\varepsilon\) accordingly larger. If the conditions are not satisfied for two function values, then the function is not continuous at that point. Thus, continuity is equivalent to uniform continuity, by choosing the largest \(\varepsilon\) from all admissible infinitesimal values. It is also easy to show that continuity is equivalent to Hölder continuity, provided that we allow infinite real constants. The same is true for uniform convergence, since we can simply choose the maximum of the indices satisfying each argument as the index that satisfies everything, and \(\acute{\omega}\) is sufficient in every case. If this is not true for a given argument, then pointwise convergence also fails. Thus, uniform convergence is equivalent to pointwise convergence, by choosing the largest of all admissible infinitesimal values.

Remark: The definition of the real numbers via Dedekind cuts is therefore just as unsuitable as its definition via equivalence classes of rational Cauchy sequences (see p. 29 ff.). Conventional differentiation and integration lose the ability to distinguish between transcendental and algebraic in the conventional process of taking limits. This is problematic, for example when we wish to exactly determine the roots of a polynomial. The above remarks illustrate why conventional analysis cannot be preserved in its existing form. Therefore, we adopt here a different (more precise) approach.

Examples (cf. p. 540 - 543 with \(n \in {}^{\omega}\mathbb{N}^{*}\) and \(x \in [0, 1]\) in each case):

1. The sequence \({f}_{n}(x) = sin(nx)/\sqrt{n}\) does not tend to \(f(x) = 0\) as \(n \rightarrow \acute{\omega}\), but instead to \(f(x) = sin(\acute{\omega}x)/\sqrt{\acute{\omega}}\) with (continuous) derivative \(f'(x) = cos(\acute{\omega}x) \sqrt{\acute{\omega}}\) instead of \(f'(x) = 0\).

2. The sequence \({f}_{n}(x) = x - x^{n}/n\) does not tend to \(f(x) = x\) as \(n \rightarrow \acute{\omega}\), but instead to \(f(x) = x - {x}^{\acute{\omega}}/\acute{\omega}\) with (continuous) derivative \(f'(x) = 1 - {x}^{\acute{\omega}-1}\) instead of \(f'(x) = 1\). Conventionally, \({f}_{n}(x) = 1 - {x}^{n-1}\) is discontinuous at the point \(x = 1\).

3. The sequence \({f}_{n}(x) = ({n}^{2}/2 - |{n}^{3}(x - 1/(2n))|)(1 - sgn(x - 1/n))\) (or alternatively, expressed in terms of continuously differentiable functions\[{{f}_{n}}(x)=\left\{ \begin{matrix} 2{{n}^{3}}x \\ 2{{n}^{2}}-2{{n}^{3}}x \\ 0 \\ \end{matrix}\begin{matrix} \,\,for\,x\in \left[ 0,\tfrac{1}{2n} \right] \\ \,\,for\,x\in \left] \tfrac{1}{2n},\tfrac{1}{n} \right] \\ for\,x\in \left] \tfrac{1}{n},1 \right] \\ \end{matrix} \right.\,\]does always not tend to 0 as \(n \rightarrow \acute{\omega}\), but instead tends to different values depending on the value of \(x\) (replace \(n\) by \(\acute{\omega}\) in \({f}_{n}(x))\). Furthermore, we have that\[\int\limits_{x \in [0,1[} {{f_n}dx} = \frac{n}{2}\]and\[\int\limits_{x\in [0,1[}{fdx}=\frac{\acute{\omega}}{2}\]instead of\[\int\limits_{x \in [0,1[} {fdx} = 0\]supposedly because \(f(x) = 0\).

4. The sequence \({f}_{n}(x) = (n/2 - |{n}^{2}(x - 1/(2n))|)(1 - sgn(x - 1/n))\) (or alternatively, expressed in terms of continuously differentiable functions)\[{{f}_{n}}(x)=\left\{ \begin{matrix} 2{{n}^{2}}x \\ 2n-2{{n}^{2}}x \\ 0 \\ \end{matrix}\begin{matrix} \,\,for\,x\in \left[ 0,\tfrac{1}{2n} \right] \\ \,\,for\,x\in \left] \tfrac{1}{2n},\tfrac{1}{n} \right] \\ for\,x\in \left] \tfrac{1}{n},1 \right] \\ \end{matrix} \right.\,\]does not always tend to 0 as \(n \rightarrow \acute{\omega}\), but instead tends to different values depending on the value of \(x\) (replace \(n\) by \(\acute{\omega}\) in \({f}_{n}(x)\)). Furthermore, we have that\[\int\limits_{x \in [0,1[} {{f_n}dx} = \int\limits_{x \in [0,1[} {fdx} = {\textstyle{1 \over 2}}\]instead of\[\int\limits_{x \in [0,1[} {fdx} = 0\]supposedly because \(f(x) = 0\).

5. The sequence \({f}_{n}(x) = nx(1 - x)^{n}\) does not tend to \(f(x) = 0\) as \(n \rightarrow \acute{\omega}\), but instead to the continuous function \( f(x) = {\acute{\omega}x(1 - x)}^{\acute{\omega}}\), and takes the value \(1/e\) when \(x = 1/\acute{\omega}\).

These five examples illustrate the superiority of nonstandard analysis and the strength of using infinitesimal and infinite values.

First fundamental theorem of exact differential and integral calculus for line integrals: The function\[F(z)=\int\limits_{\gamma }{f(\zeta )dB\zeta }\]where \(\gamma: [c, x[C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow {}^{(\omega)}\mathbb{K}, c \in [a, b[C\), and choosing \(\curvearrowright B \gamma(x) = \gamma(\curvearrowright D x)\) is exactly \(B\)-differentiable, and for all \(x \in [a, b[C\) and \(z = \gamma(x)\)\[F' \curvearrowright B(z) = f(z).\]Proof:\[dB(F(z))=\int\limits_{t\in [d,x]C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}-\int\limits_{t\in [d,x[C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{x}{f(\gamma (t))\frac{\gamma (\curvearrowright Dt)-\gamma (t)}{\curvearrowright Dt-t}dDt}=f(\gamma (x)){{{\gamma }'}_{\curvearrowright }}D(x)dDx=\,f(\gamma (x))(\curvearrowright B\gamma (x)-\gamma (x))=f(z)dBz.\square\]Second fundamental theorem of exact differential and integral calculus for line integrals: According to the conditions from above, we have with \(\gamma: [a, b[C \rightarrow {}^{(\omega)}\mathbb{K}\) that\[ F(\gamma (b))-F(\gamma (a))=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.\]Proof:\[F(\gamma (b))-F(\gamma (a))=\sum\limits_{t\in [a,b[C}{F(\curvearrowright B\,\gamma (t))}-F(\gamma (t))=\sum\limits_{t\in [a,b[C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t))(\curvearrowright B\,\gamma (t)-\gamma (t))}=\int\limits_{t\in [a,b[C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.\square\]Corollary: For a closed path \(\gamma\), we have with the conditions above that\[\oint\limits_{\gamma }{f(\zeta )dB\zeta :=}\int\limits_{\gamma }{f(\zeta )dB\zeta }=0,\]whenever \(f\) has an antiderivative \(F\) on \(\gamma.\square\)

Remark: The conventionally real case of both fundamental theorems may be established analogously. Given \(u, v \in [a, b[C, u \ne v\) and \(\gamma(u) = \gamma(v)\), it may be the case that \(\curvearrowright B \gamma(u) \ne \; \curvearrowright B \gamma(v)\). It should be noted that neither the integral nor the derivative is assumed to be continuous. Actual integration (as the inverse operation to differentiation) only makes sense for continuous functions if we wish to go beyond simple summation. However, if we can express the function values in the form of finitely many continuous functions whose antiderivatives may be computed in finite time, integrals may be calculated even for discontinuous functions, if necessary by appropriately applying the Euler-Maclaurin summation formula and other simplification techniques.

Definition: The tightened (right) exact B-integral according to the trapezoidal rule is defined by\[\int\limits_{z\in A}^{=}{f(z)dBz:=\sum\limits_{z\in A}{\frac{(f(z)+f(\curvearrowright B\,z))}{2}(\curvearrowright B\,z-z)}}.\]The tightened (right) exact B-integral according to the midpoint rule - assuming that \((z + \curvearrowright B z)/2\) exists - is defined by\[\int\limits_{z\in A}^{\doteq }{f(z)dBz:=\sum\limits_{z\in A}{f\left( \frac{z\,+\curvearrowright Bz}{2} \right)(\curvearrowright B\,z-z)}}.\]Remark: Since this tightened exact \(B\)-integral is clearly independent of the direction, it may be (implicitly) used to justify theorems that cancel integral values in opposite directions, such as Green's theorem (see below). In the first fundamental theorem, the derivative \(dB(F(z))/dBz\) can be tightened to the arithmetic mean \((f(z) + f(\curvearrowright B z))/2\) resp. \((f(z + \curvearrowright B z)/2)\), and similarly, in the second fundamental theorem, \(F(\gamma(b)) - F(\gamma(a))\) can be tightened to \((F(\gamma(b)) + F(\curvearrowleft B \gamma(b)))/2 - (F(\gamma(a)) + F(\curvearrowright B \gamma(a)))/2\) resp. \(F((\gamma(b) + \curvearrowleft B \gamma(b))/2) - F((\gamma(a) + \curvearrowright B \gamma(a))/2)\), which yields approximately the original results when \(f\) and \(F\) are sufficiently \(\alpha\)-continuous at the boundary. Since the exact \(B\)-integral is defined according to a rectangular rule, we can estimate the corresponding overall error relative to exact integration (see the literature on Numerical Mathematics).

Leibniz' differentiation rule: For \(f: {}^{(\omega)}\mathbb{K}^{n+1} \rightarrow {}^{(\omega)}\mathbb{K}, a, b: {}^{(\omega)}\mathbb{K}^{n} \rightarrow {}^{(\omega)}\mathbb{K}, \curvearrowright B x := {(s, {x}_{2}, ..., {x}_{n})}^{T}\), and \(s \in {}^{(\omega)}\mathbb{K} \setminus \{{x}_{1}\}\), choosing \(\curvearrowright D a(x) = a(\curvearrowright B x)\) and \(\curvearrowright D b(x) = b(\curvearrowright B x)\), we have that\[\frac{\partial }{\partial {{x}_{1}}}\left( \int\limits_{a(x)}^{b(x)}{f(x,t)dDt} \right)=\int\limits_{a(x)}^{b(x)}{\frac{\partial f(x,t)}{\partial {{x}_{1}}}dDt}+\frac{\partial b(x)}{\partial {{x}_{1}}}f(\curvearrowright Bx,b(x))-\frac{\partial a(x)}{\partial {{x}_{1}}}f(\curvearrowright Bx,a(x)).\]Proof:\[\begin{aligned}\frac{\partial }{\partial {{x}_{1}}}\left( \int\limits_{a(x)}^{b(x)}{f(x,t)dDt} \right) &={\left( \int\limits_{a(\curvearrowright Bx)}^{b(\curvearrowright Bx)}{f(\curvearrowright Bx,t)dDt}-\int\limits_{a(x)}^{b(x)}{f(x,t)dDt} \right)}/{\partial {{x}_{1}}}\;={\left( \int\limits_{a(x)}^{b(x)}{(f(\curvearrowright Bx,t)-f(x,t))dDt}+\int\limits_{b(x)}^{b(\curvearrowright Bx)}{f(\curvearrowright Bx,t)dDt}-\int\limits_{a(x)}^{a(\curvearrowright Bx)}{f(\curvearrowright Bx,t)dDt} \right)}/{\partial {{x}_{1}}}\; \\ &=\int\limits_{a(x)}^{b(x)}{\frac{\partial f(x,t)}{\partial {{x}_{1}}}dDt}+\frac{\partial b(x)}{\partial {{x}_{1}}}f(\curvearrowright Bx,b(x))-\frac{\partial a(x)}{\partial {{x}_{1}}}f(\curvearrowright Bx,a(x)).\square\end{aligned}\]Remark: We are integrating in the complex plane over a path whose start and end points are the limits of integration. If \(\curvearrowright D a(x) \ne a(\curvearrowright B x)\), then the final summand must be multiplied by \((\curvearrowright D a(x) - a(x))/(a(\curvearrowright B x) - a(x))\), and if \(\curvearrowright D b(x) \ne b(\curvearrowright B x)\), then the penultimate summand must be multiplied by \((\curvearrowright D b(x) - b(x))/(b(\curvearrowright B x) - b(x))\).

Intermediate value theorem: Let \(f: [a, b] \rightarrow {}^{(\omega)}\mathbb{R}\) be \(\alpha\)-continuous on \([a, b]\). Then \(f(x)\) takes every value between min \(f(x)\) and max \(f(x)\) to a precision of \(< \alpha\) as \(x\) ranges over \(\in [a, b]\). If \(f\) is continuous on \({}^{\omega}\mathbb{R}\), then it takes every value of \({}^{c}\mathbb{R}\) between min \(f(x)\) and max \(f(x)\).

Proof: Between min \(f(x)\) and max \(f(x)\), there is a gapless chain of overlapping \(\alpha\)-neighbourhoods centred around each \(f(x)\), by \(\alpha\)-continuity of \(f\). The second part of the claim follows from the fact that a deviation \(|f(\curvearrowright x) - f(x)| < 1/c\) or \(|f(x) - f(\curvearrowleft x)| < 1/c\) in \({}^{c}\mathbb{R}\) is smaller than the conventionally maximal admissible resolution.\(\square\)

Definition: The derivative of a function \(f: A \rightarrow {}^{(\omega)}\mathbb{R}\) where \(A \subseteq {}^{(\omega)}\mathbb{R}\), is defined to be zero if and only if 0 lies in the interval defined by the boundaries of the left and right exact derivatives.

Example: The (2\(d0\))-continuous function \(f: {}^{(\omega)}\mathbb{R} \rightarrow \{0, d0\}\) defined by\[f(x):=\left( \frac{{{(-1)}^{x/d0}}+1}{2} \right)d0\]consists of only the local minima 0 and the local maxima \(d0\), and only has the (left and right) exact derivatives \(\pm 1\).

Definition: Let \(f: A \rightarrow {}^{(\omega)}\mathbb{K}\) for \(A \subseteq {}^{(\omega)}\mathbb{K}\). We say that\[\frac{d_{\curvearrowright B\,z}^{2}Bf(z)}{{{(d\curvearrowright B\,z)}^{2}}}:=\frac{f(\curvearrowright B(\curvearrowright B\,z))-2f(\curvearrowright B\,z)+f(z)}{{{(d\curvearrowright B\,z)}^{2}}}\]is the second derivative of \(f\) at \(z \in A\) in the direction \(\curvearrowright B z\).

Remark: Higher derivatives are defined analogously. Every number \({k}_{j} \in {}^{\omega}\mathbb{N}\) for \(j \in {}^{\omega}\mathbb{N}^{*}\) of derivatives is written as an exponent after the \(j\)-th variable we are differentiating. If \(j \ge 2\), the derivatives are called partial and we replace \(d\) by \(\partial\). The exponent to be specified in the numerator is the sum of all \({k}_{j}\). Taylor series only make sense for \(\acute{\omega}\)-times \(\alpha\)-continuously differentiable functions, due to approximating and convergence-related behaviour.

Exchange theorem: The result of multiple partial derivatives of a function \(f: A \rightarrow {}^{(\omega)}\mathbb{K}\) is independent of the order of differentiation, provided that variables are only evaluated and limits are only computed at the end, if applicable (principle of latest substitution).

Proof: The derivative is uniquely determined: This is clear up to the second derivative, and the result follows by (transfinite) induction for higher-order derivatives.\(\square\)

Example: Let \(f: {}^{\omega}\mathbb{R}^{2} \rightarrow {}^{\omega}\mathbb{R}\) be defined as \(f(0, 0) = 0\) and \(f(x, y) = {xy}^{3}/({x}^{2} + {y}^{2})\) otherwise. Then:\[\frac{{{\partial ^2}f}}{{\partial x\partial y}} = \frac{{{y^6} + 6{x^2}{y^4} - 3{x^4}{y^2}}}{{{{({x^2} + {y^2})}^3}}} = \frac{{{\partial ^2}f}}{{\partial y\partial x}}\]with value \(\frac{1}{2}\) at the point (0, 0), even though the equation\[\frac{{\partial f}}{{\partial x}} = \frac{{{y^5} - {x^2}{y^3}}}{{{{({x^2} + {y^2})}^2}}} \ne \frac{{x{y^4} + 3{x^3}{y^2}}}{{{{({x^2} + {y^2})}^2}}} = \frac{{\partial f}}{{\partial y}}\]is equal to \(y\) on the left for \(x = 0\) and 0 on the right for \(y = 0\). Partially differentiating the left-hand side with respect to \(y\) gives \(1 \ne 0\), which is the partial derivative of the right-hand side with respect to \(x\).

Theorem: Splitting \(F: A \rightarrow {}^{(\omega)}\mathbb{C}\) into real and imaginary parts \(F(z) := U(z) + i V(z) := f(x, y) := u(x, y) + i v(x, y)\), and given infinitesimal \(h = |dBx| = |dBy|, h\)-homogeneous \(A \subseteq {}^{(\omega)}\mathbb{C}\), with the neighbourhood relation \(B \subseteq {A}^{2}\) for every \(z = x + i y \in A\) is holomorphic and\[r(h):=\frac{{\partial{}^{2}}Bf(x,y)}{\partial Bx\partial By\,}h\]is infinitesimal if and only if the Cauchy-Riemann partial differential equations\[\frac{{\partial Bu}}{{\partial Bx}} = \frac{{\partial Bv}}{{\partial By}},\,\,\frac{{\partial Bv}}{{\partial Bx}} = - \frac{{\partial Bu}}{{\partial By}},\]are satisfied by \(B\) in both the \(\curvearrowright\) direction and the \(\curvearrowleft\) direction.

Proof: Since we have\[\begin{aligned}F'B(z) &= \frac{{F(z \pm \partial Bx) - F(z)}}{{\pm \partial Bx}} = \frac{{F(z \pm i\partial By) - F(z)}}{{\pm i\partial By}} = \frac{{F(z + dBz) - F(z)}}{{dBz}} = \frac{{\partial Bu}}{{\partial Bx}} + i\frac{{\partial Bv}}{{\partial Bx}} = \frac{{\partial Bv}}{{\partial By}} - i\frac{{\partial Bu}}{{\partial By}} = \frac{{u(x \pm \partial Bx,y) + i\,v(x \pm \partial Bx,y) - u(x,y) - i\,v(x,y)}}{{\pm \partial Bx}} \\ &= \frac{{u(x,y \pm \partial By) + i\,v(x,y \pm \partial By) - u(x,y) - i\,v(x,y)}}{{\pm i\partial By}} = \frac{{\partial Bf}}{{\partial Bx}} = - i\frac{{\partial Bf}}{{\partial By}} = \frac{1}{2}\left( {\frac{{\partial Bf}}{{\partial Bx}} - i\frac{{\partial Bf}}{{\partial By}}} \right) = \frac{{\partial BF}}{{\partial Bz}}\end{aligned}\]and \(dBz = dBx + i dBy\) for every derivative defined on \(A\), and since\[\begin{aligned}&u(\curvearrowright Bx,y)-u(x,y)+u(x,\curvearrowright By)-u(x,y)+u(\curvearrowright Bx,\curvearrowright By)-u(\curvearrowright Bx,y)-u(x,\curvearrowright By)+u(x,y) =u(\curvearrowright Bx,\curvearrowright By)-u(x,y) \\ &=\frac{\partial Bu(x,y)}{\partial Bx}dBx+\frac{\partial Bu(x,y)}{\partial By}dBy+\frac{\partial Bu(\curvearrowright Bx,y)}{\partial By}dBy-\frac{\partial Bu(x,y)}{\partial By}dBy =\frac{\partial Bu(x,y)}{\partial Bx}dBx+\frac{\partial Bu(x,y)}{\partial By}dBy+\frac{{{\partial}^{2}}Bu(x,y)}{\partial Bx\partial By}dBxdBy=dBU(z)\end{aligned}\]
as well as the analogous formulas for \(v\) and in the \(\curvearrowleft\) direction, we have that\[F'B(z)\,dBz = dBF(z) = dBU(z) + i\,dBV(z) = \,\left( {\begin{array}{*{20}{c}}{\frac{{\partial Bu}}{{\partial Bx}}}{\,\frac{{\partial Bu}}{{\partial By}}}\\{i\,\frac{{\partial Bv}}{{\partial Bx}}}{i\frac{{\partial Bv}}{{\partial By}}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{dBx}\\{dBy}\end{array}} \right) + \frac{{{\partial ^2}Bf(x,y)}}{{\partial Bx\partial By}}dBxdBy.\square\]The assumptions allow us to neglect the final summand, and so the claim follows.\(\square\)

Remark: The final summand may in particular be neglected whenever \(f\) is continuous. The equations for \(F'B(z)\) imply the following necessary and sufficient condition for \(F\) to be holomorphic:\[F'B(\bar z) = \frac{{\partial Bf}}{{\partial Bx}} = i\frac{{\partial Bf}}{{\partial By}} = \frac{1}{2}\left( {\frac{{\partial Bf}}{{\partial Bx}} + i\frac{{\partial Bf}}{{\partial By}}} \right) = \frac{{\partial BF}}{{\partial B\bar z}} = 0.\]Definition: When integrating identical paths in opposite positive and negative directions, we adopt the counter-directional rule for integrals, stating that when following the path in the negative direction, we must choose the function value of the successor of the argument if the function is too discontinuous, implying that the integral sums to 0 over both directions.

Remark: We require this convention in order to ensure that integrals that we expect to sum to zero do in fact do so. Without it, they could potentially have a (significantly) different value.

Counter-directional theorem: If the path \(\gamma: [a, b[C \rightarrow V\) with \(C \subseteq \mathbb{R}\) passes the edges of every \(n\)-cube of side length \(d0\) in the \(n\)-volume \(V \subseteq {}^{(\omega)}\mathbb{R}^{n}\) with \(n \in \mathbb{N}_{\ge 2}\) exactly once, where the opposite edges in all two-dimensional faces of every \(n\)-cube are traversed in reverse direction, but uniformly, then, for \(D \subseteq \mathbb{R}^{2}, B \subseteq {V}^{2}, f = ({f}_{1}, ..., {f}_{n}): V \rightarrow {}^{(\omega)}\mathbb{R}^{n}, \gamma(t) = x, \gamma(\curvearrowright D t) = \curvearrowright B x\) and \({V}_{\curvearrowright } := \{\curvearrowright B x \in V: x \in V, \curvearrowright B x \ne \curvearrowleft B x\}\), we have that\[\int\limits_{t \in [a,b[C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)dDt}=\int\limits_{\begin{smallmatrix} (x,\curvearrowright B\,x) \\ \in V\times {{V}_{\curvearrowright}} \end{smallmatrix}}{f(x)dBx}=\int\limits_{\begin{smallmatrix} t \in [a,b[C, \\ \gamma | {\partial{}^{n-1}} V \end{smallmatrix}}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)dDt}.\]Proof: If we consider two arbitrary squares with common edge of length \(d0\) included in one plane, then only the edges of \(V\times{V}_{\curvearrowright}\) are not passed in both directions for the same function value. They all, and thus the path to be passed, are exactly contained in \({\partial}^{n-1}V.\square\)

Remark: It is not difficult to transfer both definition and theorem to the complex numbers.

Green's theorem: Given neighbourhood relations \(B \subseteq {A}^{2}\) for some simply connected \(h\)-set \(A \subseteq {}^{(\omega)}\mathbb{R}^{2}\), infinitesimal \(h = |dBx|= |dBy| = |\curvearrowright B \gamma(t) - \gamma(t)| = \mathcal{O}({\omega}^{-m})\), sufficiently large \(m \in \mathbb{N}^{*}, (x, y) \in A, {A}^{-} := {(x, y) \in A : (x + h, y + h) \in A}\), and a simply closed path \(\gamma: [a, b[\rightarrow \partial A\) followed anticlockwise, choosing \(\curvearrowright B \gamma(t) = \gamma(\curvearrowright D t)\) for \(t \in [a, b[, D \subseteq {[a, b]}^{2}\), the following equation holds for sufficiently \(\alpha\)-continuous functions \(u, v: A \rightarrow \mathbb{R}\) with (not necessarily continuous) partial derivatives \(\partial Bu/\partial Bx, \partial Bu/\partial By, \partial Bv/\partial Bx\) and \(\partial Bv/\partial By\):\[\int\limits_{\gamma }{(u\,dBx+v\,dBy)}=\int\limits_{(x,y)\in {{A}^{-}}}{\left( \frac{\partial Bv}{\partial Bx}-\frac{\partial Bu}{\partial By} \right)dB(x,y)}.\]Proof: We will prove the case \(A := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : \partial A \rightarrow {}^{(\omega)}\mathbb{R}\) wlog, since the proof is analogous for each case rotated by 90°, and every simply connected \(h\)-set is a union of such sets. We will simply show that\[\int\limits_{\gamma }{u\,dBx}=-\int\limits_{(x,y)\in {{A}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}\]since the other relation may be shown analogously. Since the regions of \(\gamma\) where \(dBx = 0\) do not contribute to the integral, for negligibly small \(t := h(u(s, g(s)) - u(r, g(r)))\), we have that\[-\int\limits_{\gamma }{u\,dBx}-t=\int\limits_{r}^{s}{u(x,g(x))dBx}-\int\limits_{r}^{s}{u(x,f(x))dBx}=\int\limits_{r}^{s}{\int\limits_{f(x)}^{g(x)}{\frac{\partial Bu}{\partial By}}dBydBx}=\int\limits_{(x,y)\in {{A}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}.\square\]Remark: The choice of \(m\) depends on the required number of sets of the type specified in the above proof, the union of which yields the simply connected \(h\)-set.

Goursat's integral lemma: If \(f\) is holomorphic on a triangle \(\Delta \subseteq {}^{(\omega)}\mathbb{C}\) but does not have an antiderivative on \(\Delta\), then\[I:=\int\limits_{\partial \Delta }{f(\zeta )dB\zeta }=0.\]Refutation of conventional proofs based on estimation by means of a complete triangulation: The direction in which \(\partial\Delta\) is traversed is irrelevant. If \(\Delta\) is fully triangulated, then wlog every minimal triangle \({\Delta}_{s} \subseteq \Delta\) must either satisfy\[{I_s}: = \int\limits_{\partial {\Delta _s}} {f(\zeta )dB\zeta } = f({z_1})({z_2} - {z_1}) + f({z_2})({z_3} - {z_2}) + f({z_1})({z_1} - {z_3}) = (f({z_1}) - f({z_2}))({z_2} - {z_3}) = 0\]or\[\begin{aligned}\int\limits_{\partial {\Delta _s}} {f(\zeta )dB\zeta } &= f({z_1})({z_2} - {z_1}) + f({z_2})({z_3} - {z_2}) + f({z_3})({z_1} - {z_3}) = (f({z_1}) - f({z_2})){z_2} + (f({z_2}) - f({z_3})){z_3} + (f({z_3}) - f({z_1})){z_1}\\ &= f'({z_2})\left( {({z_1} - {z_2}){z_2} - ({z_3} - {z_2}){z_3} + ({z_3} - {z_2}){z_1} - ({z_1} - {z_2}){z_1}} \right) = f'({z_2})\left( {({z_3} - {z_2})({z_1} - {z_3}) - {{({z_1} - {z_2})}^2}} \right) = 0\end{aligned}\]where \({z}_{1}, {z}_{2}\), and \({z}_{3}\) represent the vertices of \({\Delta}_{s}\). By holomorphicity and cyclic permutations, this can only happen for \(f({z}_{1}) = f({z}_{2}) = f({z}_{3})\). If we consider every adjacent triangle to \(\Delta\), we deduce that \(f\) must be constant, which contradicts the assumptions. This is because the term in large brackets is translation-invariant, since otherwise we can set \({z}_{3} := 0\) wlog, making this term 0, in which case \({z}_{1} = {z}_{2}(1 \pm i\sqrt{3})/2\) and \(|{z}_{1}| = |{z}_{2}| = |{z}_{1} - {z}_{2}|\). However, since every horizontal and vertical line is homogeneous on \({}^{(\omega)}\mathbb{C}\), this cannot happen, otherwise the corresponding sub-triangle would be equilateral and not isosceles and right-angled. Therefore, in both cases, \(|{I}_{s}|\) must be at least \(|f'({z}_{2}) \mathcal{O}({d0}^{2})|\), by selecting the vertices 0, \(|d0|\) and \(i|d0|\) wlog. Denoting the perimeter of a triangle by \(L\), then we have that \(|I| \le {4}^{m} |{I}_{s}|\) for an infinite natural number \(m\), and also \({2}^{m} = L(\partial\Delta)/|\mathcal{O}({d0}^{2})|\) since \(L(\partial\Delta) = {2}^{m} L(\partial{\Delta}_{s})\) and \(L(\partial{\Delta}_{s}) = |\mathcal{O}({d0}^{2})|\). Therefore, we have that \(|I| \le |f'({z}_{2}) {L(\partial\Delta)}^{2}/\mathcal{O}({d0}^{2})|\), causing the desired estimate \(|I| \le |\mathcal{O}(dB\zeta)|\) to fail, for example if \(|f'({z}_{2}) {L(\partial\Delta)}^{2}|\) is larger than \(|\mathcal{O}({d0}^{2})|.\square\)

Cauchy's integral theorem: Given the neighbourhood relations \(B \subseteq {A}^{2}\) and \(D \subseteq [a, b]\) for some simply connected \(h\)-set \(A \subseteq {}^{\omega}\mathbb{C}\), infinitesimal \(h\), a holomorphic function \(f: A \rightarrow {}^{\omega}\mathbb{C}\) and a closed path \(\gamma: [a, b[\rightarrow \partial A\), choosing \(\curvearrowright B \gamma(t) = \gamma(\curvearrowright D t)\) for \(t \in [a, b[\), we have that\[\int\limits_{\gamma }{f(z)dBz}=0.\]Proof: By the Cauchy-Riemann partial differential equations and Green's theorem, with \(x := Re z, y := Im z, u := Re f, v := Im f\) and \({A}^{-} := \{z \in A : z + h + ih \in A\}\), we have that\[\int\limits_{\gamma }{f(z)dBz}=\int\limits_{\gamma }{\left( u+iv \right)\left( dBx+idBy \right)}=\int\limits_{z\in {{A}^{-}}}{\left( i\left( \frac{\partial Bu}{\partial Bx}-\frac{\partial Bv}{\partial By} \right)-\left( \frac{\partial Bv}{\partial Bx}+\frac{\partial Bu}{\partial By} \right) \right)dB(x,y)}=0.\square\]Fundamental theorem of algebra: For every non-constant polynomial \(p \in {}^{(\omega)}\mathbb{C}\), there exists some \(z \in {}^{(\omega)}\mathbb{C}\) such that \(p(z) = 0\).

Indirect proof: By performing an affine substitution of variables, we can reduce to the case \(1/p(0) \ne \mathcal{O}(d0)\). Suppose that \(p(z) \ne 0\) for all \(z \in {}^{(\omega)}\mathbb{C}\). Since \(f(z) := 1/p(z)\) is holomorphic, we have that \(f(1/d0) = \mathcal{O}(d0)\). By the mean value inequality \(|f(0)| \le {|f|}_{\gamma}\) (see [473], p. 160) for \(\gamma = \partial\mathbb{B}_{r}(0)\) and arbitrary \(r \in {}^{(\omega)}\mathbb{R}_{>0}\), and hence \(f(0) = \mathcal{O}(d0)\), which is a contradiction.\(\square\)

Remark: The differently in \({\mathbb{B}}_{\omega}(0) \subset {}^{\omega}\mathbb{C}\) entire functions \(f(z) = \sum\limits_{k=1}^{\acute{\omega}}{{{z}^{k}}}/{{{\omega{}^{k+1}}}}\) and \(g(z) = z/\omega\) give counterexamples to Liouville's (generalised) theorem and Picard's little theorem because of \(|f(z)| < 1\) and \(|g(z)| \le 1\). The function \(f(1/z)\) for \(z \in {\mathbb{B}}_{\omega}(0)^{*}\) gives a counterexample to Picard's great theorem. The function \(b(z) := z/c\) for \(z \in {\mathbb{B}}_{c}(0) \subset {}^{c}\mathbb{C}\) maps the simply connected \({\mathbb{B}}_{c}(0)\) holomorphicly, but not necessarily injectively or surjectively to \(\mathbb{D}\) disproving the Riemann mapping theorem and analogously the (generalised) Poincaré conjecture.

Remark: If we identify \(1/\omega\) with 0, the main theorem of Cauchy's theory of functions is true and can be proven according to Dixon (as in [473], p. 228 f.), especially as the limit mentioned there shall be 0 resp. \(1/r\) tends to 0 for \(r \in {}^{\omega}\mathbb{R}_{>0}\) tending to \(\omega\).

Definition: A point \({z}_{0} \in M \subseteq {}^{(\omega)}\mathbb{C}^{n}\) or belonging to a sequence \(({a}_{k})\) for \({a}_{k} \in {}^{(\omega)}\mathbb{C}^{n}\) and an (infinite) natural number \(k\) is called a (proper) \(\alpha\)-accumulation point of \(M\) or of the sequence, if the ball \(\mathbb{B}_{\alpha}({z}_{0}) \subseteq {}^{(\omega)}\mathbb{C}^{n}\) with centre \({z}_{0}\) and infinitesimal radius \(\alpha\) contains infinitely many points from \(M\) or infinitely many pairwise distinct members of the sequence. If the claim holds for \(\alpha = 1/\omega\), the \(\alpha\)-accumulation point is simply called an accumulation point.

Remark: Let \(p(z) = \prod\limits_{k=0}^{\acute{\omega}}{\left( z-{{d}_{k}} \right)}\) with \(z \in {}^{\omega}\mathbb{C}\) be an infinite product with pairwise distinct zeros \({d}_{k} \in \mathbb{B}_{1/\acute{\omega}}(0) \subset \mathbb{D}\), chosen in such a way that \(|f({d}_{k})| < 1/\acute{\omega}\) for a function \(f\) holomorphic on a region \(G \subseteq \mathbb{C}\) and that \(f(0) = 0\). Suppose that \(G\) contains \(\mathbb{B}_{1/\acute{\omega}}(0)\) completely. This can always be achieved by means of coordinate transformations provided that \(G\) is sufficiently "large".

Then the coincidence set \(\{\zeta \in G : f(\zeta) = g(\zeta)\}\) of the function \(g(z) := f(z) + p(z)\), which is also holomorphic on \(G\), contains an accumulation point at 0, and \(f \ne g\), contradicting the statement of the identity theorem. Examples of such \(f\) include functions with a zero at 0 that are restricted to \(\mathbb{B}_{1/\acute{\omega}}(0)\) and simultaneously holomorphic on \(G\). Since \(p(z)\) can take every conventional complex number, the deviation between \(f\) and \(g\) is non-negligible.

The identity theorem is also contradicted by the fact that all derivatives \({u}^{(n)}({z}_{0}) = {v}^{(n)}({z}_{0})\) of two functions \(u\) and \(v\) can be equal at a point \({z}_{0} \in G\) for all \(n\), but that \(u\) and \(v\) can be significantly different further away from this local behaviour without ceasing to be holomorphic, since not every holomorphic function can be (uniquely) developed into a Taylor series due to the approximation (of differentiation) and computation with Landau symbols (see Number Theory).

Extending to \(\prod\limits_{k=0}^{\left| \mathbb{N}^{*} \right|}{\left( z-{{d}_{k}} \right)}\) allows entire functions with an infinite natural number of zeros to be constructed. The set of zeros is not necessarily discrete. Thus, the set of all functions that are holomorphic on a region \(G\) may contain zero divisors. Functions such as polynomials with \(n > 2\) pairwise distinct zeros once again give counterexamples to Picard's little theorem, since they omit at least \(n - 1\) values in \(\mathbb{C}\).

Remark: From the identity\[{{s}^{(0)}}(x):=\sum\limits_{k=0}^{n}{{{(-x)}^{k}}}=\frac{1-{{(-x)}^{n+1}}}{1+x}\]for real or complex \(x\), by differentiating, we deduce:\[{{s}^{(1)}}(x)=-\sum\limits_{k=1}^{n}{k{{(-x)}^{k-1}}}=\frac{(n+1){{(-x)}^{n}}-n{{(-x)}^{n+1}}-1}{{{(1+x)}^{2}}},\]when the modulus of \(x\), \(dx\) or \(1/dx\) have different orders of magnitude.

For sufficiently small but not excessively small \(x\), and sufficiently large but not excessively large (infinite) \(n\), this formula can be further simplified to \(-1/{(1+x)}^{2}\), and also remains valid when \(x \ge 1\) is not excessively large. By successively multiplying \({s}^{(j)}(x)\) by \(x\) for \(j \in {}^{\omega}\mathbb{N}^{*}\) and subsequently differentiating, we can derive other formulas for \({s}^{(j+1)}(x)\), providing an example of divergent series that has previously not always been correctly calculated.

For all complex \(z, \; {s}^{(0)}(z)\) holds. In the problematic special case of \(z = -1\), this follows from L'Hôpital's rule. However, if we integrate \({s}^{(0)}(-x)\) from 0 to 1 and set \(n := \acute{\omega}\), we obtain an integral expression for \(ln \, \acute{\omega} + \gamma\) in terms of Euler's constant \(\gamma\). Substituting \(y := 1 - x\), by the binomial series we obtain a series with almost exclusively infinite coefficients; if we also express \(ln \, \acute{\omega}\) as a series, we even obtain an expression for \(\gamma\).

If we illegitimately simplify the numerator of \({s}^{(0)}(z)\), we risk finding incorrect results, especially when \(|z| \ge 1\). So \({s}^{(0)}(-{e}^{i\pi})\) is e.g. 0 for odd \(n\), and 1 for even \(n\), but not \(\frac{1}{2}\).

Theorem: Using the digamma function \(\psi\), we have for \(n \in {}^{\omega}2\mathbb{N}^{*}\) and small \(d \in ]0, 1]\) that\[\zeta (n+1)=\underset{d\to 0}{\mathop{\lim }}\,\frac{-1}{n{{d}^{n}}}\sum\limits_{k=1}^{n}{\left( \gamma +\psi \left( d{{e}^{\tfrac{2k\pi i}{n}}} \right) \right)}+\mathcal{O}\left( {{d}^{n}} \right)\]and\[\zeta (n+1)=\underset{d\to 0}{\mathop{\lim }}\,\frac{1}{2n{{d}^{n}}}\sum\limits_{k=1}^{n}{\left( \psi \left( d{{e}^{\tfrac{2k\pi i}{n}}}{{i}^{\tfrac{2}{n}}} \right)-\psi \left( d{{e}^{\tfrac{2k\pi i}{n}}} \right) \right)}+\mathcal{O}\left( {{d}^{2n}} \right).\]Proof: The claim results easily via the geometric series from ([474], p. 37 - 42):\[\psi (z)+\gamma +\frac{1}{z}=\sum\limits_{k=1}^{\acute{\omega}}{\left( \frac{1}{k}-\frac{1}{k+z} \right)}=-\sum\limits_{k=1}^{\acute{\omega}}{\zeta (k+1){{(-z)}^{k}}}=\sum\limits_{k=1}^{\acute{\omega}}{\frac{z}{k\left( k+z \right)}}.\square\]Remark: We may well accelerate the slowly convergent series on the right by Euler summation.

Corollary: For \(n \in {}^{\omega}2\mathbb{N}^{*}\), we have that\[\zeta (n+1)=\frac{-{\omega{}^{n}}}{n}\sum\limits_{k=1}^{n}{\left( \gamma +\psi \left( {{{e}^{\tfrac{2k\pi i}{n}}}}/{\omega }\; \right) \right)}.\square\]Corollary: For \(z \in \mathbb{B}_{1-1/c}(0) \subset \mathbb{D}, s \in {}^{\omega}\mathbb{C}, 1/c \le Re \; s < 1 + 1/c\) and small \(d \in ]0, 1]\), we have with\[v(s,z):=\sum\limits_{k=1}^{\acute{\omega}}{\frac{z}{{{k}^{s}}(k-z)}}=\sum\limits_{k=1}^{\acute{\omega}}{\zeta (k+s){{z}^{k}}}\]and \(u(s, z) := zv(s, z)\) as well as \(n \in {}^{\omega}2\mathbb{N}^{*}\) the formulas\[\zeta (n-1+s)=\underset{d\to 0}{\mathop{\lim }}\,\frac{1}{n{{d}^{n}}}\sum\limits_{k=1}^{n}{u\left( s,d{{e}^{\tfrac{2k\pi i}{n}}} \right)}+\mathcal{O}\left( {{d}^{n}} \right)\]and\[\zeta (n+s)=\underset{d\to 0}{\mathop{\lim }}\,\frac{1}{n{{d}^{n}}}\sum\limits_{k=1}^{n}{v\left( s,d{{e}^{\tfrac{2k\pi i}{n}}} \right)}+\mathcal{O}\left( {{d}^{n}} \right).\]Proof: Since the first series is convergent, the claim follows analogously as before.\(\square\)

Remark: Adding \(p/z + qz + r\) to \(u(s, z)\) resp. \(v(s, z)\) with \(p, q, r \in {}^{\omega}\mathbb{C}\) barely matters.

Corollary: With \(a \in D \subseteq \mathbb{C}\) and \(n \in {}^{\omega}2\mathbb{N}^{*}\), we have for a function\[f(z)=\sum\limits_{k=0}^{\acute{\omega}}{\frac{{{f}^{(k)}}(a)}{k!}{{(z-a)}^{k}}}\]holomorphic in the domain \(D\) and \(g(z) := zf(z + a)\) resp. \(h(z) := f(z + a) - f(a)\) the formulas\[{{f}^{(n-1)}}(a)=\underset{d\to 0}{\mathop{\lim }}\,\frac{(n-1)!}{n{{d}^{n}}}\sum\limits_{k=1}^{n}{g\left( d{{e}^{\tfrac{2k\pi i}{n}}} \right)}+\mathcal{O}\left( {{d}^{n}} \right)\]and\[{{f}^{(n)}}(a)=\underset{d\to 0}{\mathop{\lim }}\,\frac{(n-1)!}{{{d}^{n}}}\sum\limits_{k=1}^{n}{h\left( d{{e}^{\tfrac{2k\pi i}{n}}} \right)}+\mathcal{O}\left( {{d}^{n}} \right).\]Proof: The claim follows by the above from Taylor's theorem (cf. [473], p. 165 f.).\(\square\)

Remark: We may arbitrarily increase the precision by successively inserting the terms.

© 07.03.2018 by Boris Haase

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