Preliminary remark: In the following, the definitions from the set theory are applied and let be at most m, n ∈ ^{ω}ℕ*. The integration and differentiation on arbitrary always non-empty subsets A of the sets ^{(ω)}ℂ^{n} or ^{(ω)}ℝ^{n} with arbitrary n are studied, which we combine into the sets ^{(ω)}ℍ^{n} = ^{(ω)}ℍ× ... ×^{(ω)}ℍ, where each ℍ can be both ℂ and ℝ in arbitrary order, since quaternions are not to be considered here. Especially conventionally non-measurable, inconcrete and infinite sets, as well as discontinuous functions are treated. Each element outside of the image set is replaced by the next one of the target set, and we select one of them if it is not unique. Otherwise, the mapping is not (rationally) defined. A generalisation to other sets is easily possible.

Definition: The function ||·||: V → ^{(ω)}ℝ_{>0} with the vector space V over ^{(ω)}ℍ is called *norm*, if we have for all x, y ∈ V and all λ ∈ ^{(ω)}ℍ: ||x|| = 0 ⇒ x = 0 (*definiteness*), ||λx|| = |λ| ||x|| (*homogeneity*) and ||x + y|| ≤ ||x|| + ||y|| (*triangle inequality*). The *dimension* of V as the maximal number of linearly independent vectors is denoted by dim V. The norms ||·||_{a} and ||·||_{b} are called *equivalent* if there exist finite, but not infinitesimal σ, τ ∈ ^{(ω)}ℝ_{>0} such that we have for all x ∈ V:

σ||x||_{b} ≤ ||x||_{a} ≤ τ||x||_{b}.

Theorem: Let N be the set of all norms in V. These are equivalent iff ||x||_{a}/||x||_{b} is finite, but not infinitesimal for all ||·||_{a}, ||·||_{b} ∈ N and all x ∈ V*.

Proof: With σ := min {||x||_{a}/||x||_{b}: x ∈ V*} and τ := max {||x||_{a}/||x||_{b}: x ∈ V*}, the assertion follows immediately.⃞

Remark: All norms are equivalent in ^{κ}ℍ^{n}, with n ∈ ^{ω}ℕ*, if there the definition of infinite or infinitesimal norm values is foregone. In the following, ||·|| is the Euclidean norm. A generalisation to other norms is easily possible, if they are equivalent.

Determination: In the nonstandardanalysis represented here, only h-homogeneous sets and derived ones from them are considered, whose inhomogeneity can be clearly and unambiguously described.

Definition: We describe neighbour points in A by the irreflexive symmetric *neighbourhood relation* B ⊆ A^{2}. The set ÑB(z_{0}) of all neighbours of z_{0} ∈ A (with respect to B) in A is called *neighbourhood* of z_{0}. The function γ: C → A ⊆ ℂ^{n} with a h-homogeneous C ⊆ ℝ and infinitesimal h is called *path*, if ||γ(x) – γ(y)|| is infinitesimal for neighbours x, y ∈ C and (γ(x), γ(y)) ∈ B. The neighbourhood relations in B in A are noted always as (predecessor, successor) in the form (z_{0}, ↷z_{0}) or (↶z_{0}, z_{0}), whereby ↷ and ↶ are pronounced "suc" and "pre". This applies analogously for the neighbourhood relation D ⊆ C^{2}.

Definition: Let be z_{0} ∈ A ⊆ ℍ^{n} and f: A → ^{(κ)}ℍ^{m}. In the following, the proofs with predecessors are usually omitted, since they run similarly to those with successors. Then f is called *αB-successor-continuous* in z_{0} in direction ↷B z_{0} if for infinitesimal α ∈ ^{(ω)}ℝ_{>0} applies: :

||f(↷B z_{0}) - f(z_{0})|| < α.

Generally, we write for ||x – y|| < α with x, y ∈ A x ≈_{α} y, what we pronounce "α-infinitesimally equal". If the exact modulus of α is immaterial, we omit α, too. If f is for all z_{0} and ↷B z_{0} ∈ ÑB(z_{0}) αB-successor-continuous, then we speak simply of αB-continuity. Here α is called the *degree* of continuity. If the inequality applies only for α = 1/⌊ω⌋, we speak simply of (B-successor-) continuity. The αB-predecessor-continuity results analogously.

Remark: In practice, we will determine α by estimating (after consideration of any jump discontinuities of f). If B is obvious or unimportant, it can be omitted. This is in the following always the case if B = ^{(ω)}ℍ^{2n} applies.

Example: The function f: ℝ → {±1} with f(x) = (-1)^{x/d0} is in ℝ nowhere successor-continuous, but indeed its modulus (cf. transcendental numbers). Here x/d0 is integer because of the d0-homogenity of ℝ. If we set f(x) = 1 for rational x and = -1 for irrational ones, so f(x) is, in the irrational x, partially d0- successor-continuous, contrary to the conventional perception.

Definition: For f: A → ^{(ω)}ℍ^{m},

d_{↷B z}f(z) := f(↷B z) - f(z)

is called *B-successor-differential* of f towards ↷B z in z ∈ A. If Dim A = n is true, so we can specify d↷B zf(z) d_{↷B z}f(z) by d((↷B)z_{1}, ... , (↷B)z_{n})f(z). If f is the identity, that is f(z) = z, we can write d_{↷B z}Bz instead of d_{↷B z}f(z). If A or ↷B z are obvious or unimportant, they can be omitted. The conventionally real case results analogously as above.

Remark: If the modulus of the B-successor-differential of f towards ↷B z in z ∈ A is smaller than α and infinitesimal, so f is there also αB-successor-continuous.

Definition: The m arithmetical means of all f_{k}(↷B z) of f(z) form the m *averaged normed tangential normal vectors* of m (uniquely determined) hyper planes, which yield the mn continuous partial derivatives for the Jacobian matrix of a not unconditionally continuous f. We set the hyper planes here so that they run each through f_{k}(↷B z) and the f(z) translated to 0, and minimise afterwards the modulus of their coefficients in a linear linear programme solvable very easily (cf. linear optimisation).

Proposition: For the bijectivity of the mapping f: X → X with arbitrary functions f and sets X injectivity resp. surjectivity is sufficient.

Proof: The assertion follows directly from all (inverse) images being pairwise different.⃞

Remark: The successor mapping s in ^{ω}ℕ is not of this type, since we have s: ^{ω}ℕ → ^{ω}ℕ* ∪ {|^{ω}ℕ|}.

Example of a Peano curve (translated from [739], p. 188, see bibliography): "Let the function g: ℝ → ℝ be even, periodic with period 2 and given in [0, 1] by

Obviously, g is completely defined and continuous by this specification. Let the function Φ: I = [0, 1] → ℝ^{2} be defined by

"

The function Φ is at least continuous, since the sums locally are ultimately linear functions in t, if ∞ is replaced by ⌊ω⌋. But it is a mistake to believe that in this way [0, 1] can be mapped bijectively onto [0, 1]^{2}, since e.g. the powers of four in g provide, together with the values 0 and 1 attained by g in two sub-intervals, for a so strong thinning in [0, 1]^{2} that a bijection can be no question. Limiting a proof to rational points is simply insufficient.

Definition: A point x (↷x) of a function f: A ⊆ ^{ω}ℝ → ^{ω}ℝ is called *jump discontinuity* with *jump to the right (left)* of s:= |f(↷x) – f(x)| *upward (downward)* or vice versa, if s > 1/|^{ω}ℕ*| applies.

Theorem: A monotone function f: [a, b] → ^{ω}ℝ has maximally |^{ω}ℕ||^{ω}ℤ*| - 1 jump discontinuities.

Proof: Between -|^{ω}ℕ*| and |^{ω}ℕ*|, maximally |^{ω}ℕ*||^{ω}ℤ*| jump discontinuities with jump 1/|^{ω}ℕ*| are possible, beyond that, taken together, maximally |^{ω}ℤ*| - 1 ones. If the function holds its values like a step function, except at the jump discontinuities, the assertion follows, if the remainders at the ends of the number line ℝ are properly considered.⃞

Remark: Froda's theorem is put right and stated more precisely by this one. If we put ^{κ} in front of all sets, we obtain the statement for conventional sets.

Definition: The set ∁A := X \ A, for A ⊆ X, with X as an arbitrary set, is called *complement* of A in X. If X is clear, it is omitted and ∁A is also referred to as the *exterior* of A. The set ∂A consists of all points of A that have a neighbour from ∁A and is called *(inner) boundary* of A. For sets without complement, the boundary consists of its extreme elements, if they exist. The set ℰA := ∂∁A is called *exterior boundary* of A. The set A° := A \ ∂A is called the *interior* of A. The set B_{(≤)r}(a) := {z ∈ H := ^{(ω)}ℍ^{n} : ||z - a|| ≤ r} is called *ball* with *radius* r ∈ ^{(ω)}ℝ around the *centre* a ∈ H, whose boundary is called *sphere*. For a = 0 and r = 1, we obtain the *unit sphere* with the special case of the *unit disk* đ for n = 2. The set B_{<r}(a) := {z ∈ H : ||z - a|| < r} = B_{r}(a)° is correspondingly called *inner ball*.

Remark: Sets without complement have exclusively no boundary because of missing extreme elements.

Definition: The function µ_{h}: A → ℝ_{≥0} with a h-homogeneous m-dimensional set ^{(ω)}ℂ^{n}, m ∈ ℕ*_{≤2n} and µ_{h}(A) := |A| h^{m} as well as µ_{h}(∅) = |∅| = 0 is called *exact h-measure* of A and A *h-measurable*, where dim ^{(ω)}ℂ = 2 applies.

Remark: Obviously, µ_{h}(A) is additive and uniquely determined, i.e., if A is a union of pairwise disjoint h-homogeneous sets A_{k} with k ∈ ℕ, so we have

It is moreover strictly monotone, i.e. for h-homogeneous sets A_{1}, A_{2} ⊆ ^{(ω)}ℍ^{n} with A_{1} ⊂ A_{2} applies µ_{h}(A_{1}) < µ_{h}(A_{2}). If h is not equal for all considered sets A_{k}, so we choose the minimum of all h and homogenise as described above. Thus, it measures more accurately than other measures and is optimal, since it turns out neither smaller nor greater than the individual distances of the points parallel to the coordinate axes amount, because it merely considers the neighbourhood of a point. Concepts such as σ-algebra or null sets are not needed here, since the empty set is the only null set here.

Examples: If A ⊂ [0, 1[ consists of the points that have a 1 (0) at the end of their (conventionally) real dual representation (the end, uniquely determined this way, is guaranteed by the minimax proposition of the set theory), so we have µ_{d0}(A) = ½. The real numbers still refine the conventionally real ones by dividing the conventionally real intervals still (significantly) finer. Since A is an infinite union (conventionally not countable) of single points (without neighbour points from [0, 1[ in A) and these point-sets represent Lebesgue null sets, A is not Lebesgue measurable, but exactly measurable however. Analogously, we can consider in [0, 1[ × [0, 1[ the set S of all points with end 1 (0) in their both coordinates with exact measure µ_{d0}(S) = ¼.

Remark: Also with respect to BBP series, the dual representation is suitable - or at least a power of two as base (ideally 8, cf. calculation of times).

Definition: The *partial derivative* towards ↷B z_{k} of F: A → ^{(ω)}ℍ in z = (z_{1}, ..., z_{n}) ∈ A ⊆ ^{(ω)}ℍ^{n} with k ∈ [1, n]ℕ is defined as

If we have, with the notations above,

for a function f = (f_{1}, ..., f_{n}): A → ^{(ω)}ℍ^{n} with z ∈ A ⊆ ^{(ω)}ℍ^{n} then f(z) is called *exact B-successor-derivative* F′_{↷B z} B(z) or *exact B-successor-gradient* grad_{↷B z} F(z) towards ↷B z of the *exactly B-differentiable* function F in z, if all quotients exist in ^{(ω)}ℍ. Here ∇ is the *Nabla operator*. If this applies for all z ∈ A, then F is called *exactly B-differentiable B-antiderivative* of f. In the (conventionally) (infinitely) real, left-sided and right-sided B-antiderivatives F_{l}(x) and F_{r}(x) with x ∈ ^{(ω)}ℝ can be distinguished, depending on whether we face the left-sided or right-sided B-derivative.

If A or ↷B z are obvious or unimportant, they can be omitted. The conventionally case results analogously as above, and we speak for ↷B w > w ∈ ^{(ω)}ℝ of the *right-sided exact B-derivative* F′_{r}B(w), and for ↷B w < w we do so of the *left-sided exact derivative* F′_{l}B(w). If the derivatives match towards all neighbours, we speak correspondingly of the exact derivative F′B(z) (for A = ^{κ}ℂ and n = 1, F is then conventionally regarded as *holomorphic*).

Remark: Obviously, the B-antiderivatives of a function differ from each other only by a (conventionally) (infinite) complex or (infinite) real addend. B-antiderivatives of discontinuous functions can usually only be obtained by adding them up and skilfully combining them, such of piecewise αB-continuous functions easier (for example, by reversing the rules of derivation).

Chain rule: For x ∈ A ⊆ ^{(ω)}ℝ, B ⊆ A^{2}, f: A → C ⊆ ^{(ω)}ℝ, D ⊆ C^{2}, g: C → ^{(ω)}ℝ we have, if we choose f(↷B x) = ↷D f(x),

g′_{r}B(f(x)) = g′_{r}D(f(x)) f′_{r}B(x).

Proof:

Remark: For reasons of simplicity and clarity, sets of neighbourhood relations are omitted, but to be understood. Let be furthermore f and g right-sided exactly differentiable (infinite) real functions in x ∈ A ⊆ ^{(ω)}ℝ.

Product rule: It applies

(fg)′_{r}(x) = f′_{r}(x) g(x) + f(↷ x) g′_{r}(x)= f′_{r}(x) g(↷ x) + f(x) g′_{r}(x).

Proof: We add and subtract in the numerator f(↷ x) g(x) resp. f(x) g(↷ x).⃞

Quotient rule: Do not let the denominators of the following quotients be 0. Then we have

Proof: We add and subtract in the numerator f(x) g(x) resp. f(↷ x) g(↷ x).⃞

Remark: In order that the product and quotient rule coincides precisely enough with the conventional one, the arguments and function values must belong to a smaller infinity level than 1/d0, as well as f and g must be (α-) continuous enough in x ∈ A (i.e., we can set α small enough). We can say analogous from infinitesimal arguments.

Remark: The right-sided exact derivative of the inverse function results as

f^{-1}′_{r}(y) = 1/f′_{r}(x)

from y = f(x) and the identity x = f^{-1}(f(x)), with the aid of the chain rule, for the same precision. L'Hôpital's rule is useful for (α-) continuous functions f and g and results for f(w) = g(w) = 0 with w ∈ A, as well as f(↷ w) and g(↷ w), not both simultaneously 0 from

Remark: Differentiability thus can be easily established. We can define the exact derivative in the (conventionally) (infinite) real case alternatively also everywhere there as

where the quotient is defined. This is above all convenient, if ↷B w - w = w - ↶B w and if the summarised derivatives have the same sign, and has the advantage that F′_{b}B(w) can be regarded more as "tangent slope" in the point w, especially if F is αB-continuous in w. This provides furthermore simple derivation rules, especially as the derivative value 0 is best suitable for opposite signs (see below). Otherwise we build simply the arithmetic mean of the two exact derivatives. The transfer to the conventionally complex is done analogously. On this understanding we have the

Definition: With z ∈ A ⊆ ^{(ω)}ℍ^{n} as well as where dBz resp. ↷B z and the right-hand side exist in ℍ^{n},

is called the *exact B-integral* of a *vector field* f = (f_{1}, ..., f_{n}): A → ^{(ω)}ℍ^{n} in A and f(z) is called *B-integrable*. If we have to achieve this first to remove at least one point from A, so the exact B-Integral is called *improper*. For γ: [a, b[A → ^{(ω)}ℍ^{n}, A ⊆ ℝ and f = (f_{1}, ..., f_{n}): ^{(ω)}ℍ^{n} → ^{(ω)}ℍ^{n}

with dDt > 0, ↷D t ∈ ]a, b]A, if we choose ↷B γ(t) = γ(↷D t), because of ζ = γ(t) and dBζ = γ(↷D t) - γ(t) = γ′_{↷}D(t) dDt (so especially for A = ℝ, B maximal in ^{ω}ℂ^{2} and D maximal in ^{ω}ℝ^{2}), is called the *exact B-line integral* of a vector field f along the path γ, if the right-hand integral exists in ℍ^{n}. Improper exact B-line integrals are defined analogously to the exact B-integrals, where the points may be each removed only from the interval ends of [a, b[A.

Standard estimate: For n = 1, thus we obtain, with M = max |f(z)|, on γ successively using the triangle inequality

where the last sum defines also for n > 1 the *Euclidean path length* L(γ).

Remark: The conventionally real case results analogously as above. Obviously, the exact integration is a special case of summation. The exact integral coincides on ^{(κ)}ℍ widely with conventional line integrals; f must yet not be continuous and also else the conditions are significantly weaker, in order that the line integral exists. Obviously, the exact line integral is linear and, in the conventionally real case, monotone. The art of integrating consists in correctly summarising the addends. In nonstandardanalysis, we can also specify the length of infinitely long paths.

Definition: For z ∈ A = A_{1}× ... ×A_{n} ⊆ ^{(ω)}ℍ^{n} with for each z_{k} ∈ A_{k} in each case uniquely determined neighbour ↷B_{k} z_{k} of the neighbourhood relations B_{k} ⊆ A_{k}×^{(ω)}ℍ, for all k ∈ ^{ω}ℕ*_{≤n} and B = B_{1}× ... ×B_{n}

is called the *exact B-volume integral* over a then *B-volume integrable* function f = (f_{1}, ... , f_{m}): A → ^{(ω)}ℍ^{m}, if the right-hand side exists in ^{ω}ℍ^{n}. Improper exact B-volume integrals are defined analogously to the exact B-integrals.

Remark: Obviously, we have

if dBx_{k} = h is fulfilled for all x of a h-homogeneous set A ⊆ ^{(ω)}ℝ^{n} and all k ∈ [1, n]ℕ as well as the conditions mentioned for the exact h-measure.

Remark: Analogously, the exact volume integral can be alternatively defined. But the original definitions are to handle the easiest way. If applicable, there is an appropriate Landau notation. If the result of differentiation lies outside of the domain, it should be replaced by the number lying closest to it, within the domain. If the number is not uniquely determined the result is to consist of all these numbers, or we may choose one (e.g. after a uniform rule).

Example: Let be [a, b[h[1, n]^{ω}ℤ a non-empty h-homogeneous subset of [a, b[^{ω}ℝ with B = ]a, b]h[1, n]^{ω}ℤ. For h = 1/κ and κ = -a = b – h, [a, b[h[1, n]^{ω}ℤ is comparable with ^{κ}ℝ. Let be T_{r} furthermore a right-sided B-antiderivative of a Taylor series t, not necessarily convergent in [a, b[h[1, n]^{ω}ℤ, and f(x) := t(x) + ε(-1)^{x/h} with conventionally real x and ε ≥ 1/κ. For h = 1/κ, f is nowhere continuous, and can therewith nowhere be conventionally differentiated and integrated in [a, b[h[1, n]^{ω}ℤ, but it applies exactly for all h

and

Example: The middle thirds Cantor set C has the relative measure µ_{d0}(C) = (⅔)^{⌊ω⌋}. Let the function c: [0, 1] → {0, (⅔)^{-⌊ω⌋}} be defined by c(x) = (⅔)^{-⌊ω⌋} for x ∈ C and c(x) = 0 for x ∈ [0, 1] \ C. Then it applies

Example: For the classes a + ^{ω}ℚ with a ∈ ^{ω}ℝ the equivalence relation x ~ y ⇔ x - y ∈ ^{ω}ℚ with x and y ∈ ^{ω}ℝ their representatives can be specified through the set R = [0, 1/(⌊ω⌋)[ with the measure µ_{d0}(R) = 1/⌊ω⌋. Let the function r: ^{ω}ℝ → {0, 1} be defined by r(x) = 1 for x ∈ ^{ω}ℤ + R and r(x) = 0 for x ∈ ^{ω}ℝ \ (^{ω}ℤ + R). Then it applies

Example (cf. set theory): Let A_{1} = [0, 1[ ∩ ^{ω}A_{ℝ} and the function q: A_{1} → {0, 1} be defined by q(x) = 1 for x ∈ A_{1} \ ^{ω}ℚ and q(x) = 0 for x ∈ A_{1} ∩ ^{ω}ℚ. The exact integral over q(x)dx has then in A_{1} the transcendental value

Remark: The sets C, R and ^{ω}ℚ are conventionally not measurable. Thus, the exact integral is more generally valid than Riemann, Lebesgue (-Stieltjes) integral and other integrals, since the latter exist only in conventionally measurable sets. The functions were chosen so easy only because of the clearness and may be, of course, more complicated.

Definition: A *sequence* (a_{k}) with *members* a_{k} is a map from ^{(ω)}ℤ to ^{(ω)}ℂ^{m}: k ↦ a_{k}. A *series* is a sequence (s_{k}) with m ∈ ^{(ω)}ℤ and the *partial sums*

Remark: Since sums can be arbitrarily added otherwise, because of the associative, commutative and distributive law, if is calculated correctly (with the Landau symbols), it results

Fubini's theorem: For A_{1}, A_{2} ⊆ ^{(ω)}ℍ, it applies with f: A_{1}×A_{2} → ^{(ω)}ℍ

Proof: The assertion results directly by rearranging the sums corresponding to the integrals above.⃞

Example: Because of

we obtain according to the latest-insert-principle (see below) the (improper) integral

as the case may be, and we do not have

Remark: Thus, in particular, the Riemann series theorem is invalid, since we are coerced, when summing up the positive summands to a value aimed at, to add so many negative ones, until we obtain again the original sum of the series, and vice versa. With a smaller resp. greater value than the sum of the positive resp. negative summands, the same applies, since the rest is almost annulled, and so on. Also infinity must not be dealt with arbitrarily, if we want to avoid going astray. Who moves something into infinity must not fall into the illusion that it would no longer exist.

Finiteness criterion for series: The Euclidean norm of the partial sum with the greatest index of a real series (s_{k}) for infinite (infinite) natural k and j is finite, iff it can be represented as

with finite ||a_{j} - b_{j}|| forming a monotonically nonincreasing sequence for a_{j}, b_{j} ∈ ^{(ω)}ℂ^{m}.

Proof: The assertion follows directly from the finiteness of ||a_{1} - b_{1}|| and because the addends can be added arbitrarily otherwise, sorted by size and sign, summed up or split up into sums.⃞

Example: From the alternating harmonic series follows

Remark: More interesting examples are those whose nonincreasing monotony of |a_{j} - b_{j}| cannot be as easily proven as e.g. for the divergent series

where c_{j} increases monotonically, but c_{j+1} - c_{j} is monotonically nonincreasing.

Proposition: The number s := 1 - 1/κ is an upper bound, which makes, inserted as argument into the geometric series, the sum of the latter conventionally real.

Proof: The assertion follows directly from κ (1 - 1/κ)^{⌈ω⌉} ≈ 0.⃞

Remark: With the notation of the preceding proposition and the proofs known from the literature, it follows that the sum of a power series with terms a_{n}x^{n} is conventionally real, if the radius of convergence of the latter is ≤ s/lim sup |a_{n}|^{1/n}.

Finiteness criterion (corresponds to the fundamental theorem) for products: The product

for k ∈ ℕ* and a_{k} ∈ ^{κ}ℝ_{>0} is finite, if with finite

e^{S} is maximally finite.

Proof: With the exponential series, the assertion follows directly from S < P < e^{S}.⃞

Remark: Products with a_{k} ∈ ^{κ}ℂ are finite, iff their absolute values are. Factors with absolute value < 1 are to be set off against those with absolute value > 1, for example, by considering their product of the reciprocals.

Definition: A sequence (a_{k}) with k ∈ ^{(ω)}ℕ*, a_{k} ∈ ^{(ω)}ℂ and α ∈ ]0, 1/κ] is called *α-convergent to* a ∈ ^{(ω)}ℂ, if there is a q ∈ ℕ*, so that for all a_{k} with k ≥ q and not too small difference max k - q |a_{k} - a| < α applies. The set α-A of all such a is called *α-set of limit values* of (a_{k}), the representative uniquely and appropriately determined (e.g. as last or mean value) from α-A is called *α-limit value* α-a. We speak of a *zero sequence* for especially a = 0. If the inequality only applies for α = 1/κ, so α- is omitted.

Remark: Usually, we will choose k maximal and α minimal. The conventional limit values are often no more precise than O(1/⌊ω⌋) and generally too imprecise, since they are for example (arbitrarily) algebraic (of a certain degree), or transcendental. The conventional formulation that there always can be found infinitely many or almost all members with an arbitrarily small distance to the limit value and only finitely many with a greater one, would have to be added to the definition of the conventional convergence, since it applies otherwise for every sequence, for which only the greatest index is considered as relevant (cf. [813], p. 144, see bibliography), and only then the monotonicity principle is valid (cf. l.c. p. 155).

Remark: The statement that every positive number could be represented by a completely uniquely determined infinite decimal fraction is untenable, since we can apply the proof of the irrationality of 2 to infinite decimal fractions (cf. pp. 27 f.) Moreover, all proofs are false that assert for ε ∈ ^{(ω)}ℝ_{>0} - in particular by the formulation for all conventionally real ε > 0 - that a real number ε/r with a real r ∈ ^{(ω)}ℝ_{>1} would exist, because we can set just ε := ↷0 resp. get in an infinite regress. Therefore, for the εδ-definition of the limit value (debatable existence of δ, p. 235 f.) and thus for the εδ-definition of continuity (cf. p. 215) (say, consider the real function that doubles any real value, and then it is not even uniformly continuous), ε must be limited to a certain integer multiple of ↷0.

Remark: The consideration of uniform continuity is dispensable, since we generally can set δ := ↷0 and ε correspondingly greater. If the conditions are not satisfied for two function values, the function is there also not continuous. Thus continuity is equivalent to uniform continuity, if we choose among all valid infinitesimal ε the greatest one. The equivalence to Hölder continuity can be equally easy shown; provided we permit a infinite real constant if necessary. The same is true for uniform convergence, since we can choose as the index satisfying everything the maximum of the indices that satisfy for each argument, where |ℕ*| should be sufficient in all cases. If this is not the case for one argument, there is also given no pointwise convergence. Thus uniform convergence is equivalent to pointwise convergence, if we choose among all valid infinitesimal ε the greatest one.

Remark: Since there are infinitely many algebraic numbers of higher degree between two rational numbers (see set theory), the principle of nested intervals is refuted (cf. p. 158). The definition of the real numbers by Dedekind cuts is therefore equally unsuitable as by equivalence classes of rational Cauchy sequences (cf. p. 29 ff.). The best definition is thus the homogeneous one as (hyper-) integer multiples (i.e. integer numbers that are partially not in ℤ) of ↷0. The preceding remarks make clear that the conventional analysis cannot be maintained in the existing form.

Examples (cf. pp. 540 - 543 with in each casen ∈ ^{ω}ℕ* and x ∈ [0, 1]):

1. The sequence f_{n}(x) = sin(nx)/√n does not tend for n → ⌊ω⌋ to f(x) = 0, but to f(x) = sin(⌊ω⌋x)/√⌊ω⌋ with the (continuous) derivative f′(x) = cos(⌊ω⌋x) √⌊ω⌋ instead of f′(x) = 0.

2. The sequence f_{n}(x) = x - x^{n}/n does not tend for n → ⌊ω⌋ to f(x) = x, but to f(x) = x - x^{⌊ω⌋}/⌊ω⌋ with the (continuous) derivative f′(x) = 1 - x^{⌊ω⌋-1} instead of f′(x) = 1. Conventionally f_{n}(x) = 1 - x^{n-1} is discontinuous in the point x = 1.

3. The sequence f_{n}(x) = (n^{2}/2 - |n^{3}(x - 1/(2n))|)(1 - sgn(x - 1/n)) (or expressed with continuously differentiable functions

does not generally tend for n → ⌊ω⌋ to 0, but to different values depending on x (replace n by ⌊ω⌋ in f_{n}(x)). Furthermore, we have

and

instead of

because of allegedly f(x) = 0.

4. The sequence f_{n}(x) = (n/2 - |n^{2}(x - 1/(2n))|)(1 - sgn(x - 1/n)) (or expressed with continuously differentiable functions

does not generally tend for n → ⌊ω⌋ to 0, but to different values depending on x (replace n by durch ⌊ω⌋ in f_{n}(x)). Furthermore, we have

instead of

because of allegedly f(x) = 0.

5. The sequence f_{n}(x) = nx(1 - x)^{n} does not tend for n → ⌊ω⌋ to f(x) = 0, but to the continuous function f(x) = ⌊ω⌋x(1 - x)^{⌊ω⌋} and takes for x = 1/⌊ω⌋ the value 1/e.

These five examples show here the superiority of the nonstandardanalysis, and that it is useful to deal with infinitesimal resp. infinite values.

Theorem about commuting α-limit values for the integration: Let be A ⊆ ^{(ω)}ℝ and (f_{j}) a sequence of integrable functions with (infinite) natural j and f_{j}: A → ^{(ω)}ℝ, which α_{1}-converge to the integrable function f: A → ^{(ω)}ℝ. Then applies for:

Proof:

.⃞

Remark: As long as we correctly calculate (with Landau notation), differentiation or integration and summation may be (also) interchanged in (divergent) series. The conventional procedure can, however, lead to not inconsiderable error propagations, in subsequent calculations, for example if α_{1}μ_{d0}(A) ≥ 1/ω applies. The principle of the latest possible replacement of the variables applies for all permutations, since else discrepancies can occur. From this follows directly:

Great permuting theorem: The arbitrary permutation of the order of the (feasible) same variable substitution in sequences, derivatives and integrals leads to the same result.

First fundamental theorem of exact calculus for line integrals: The function

with γ: [c, x[A → ^{(ω)}ℍ^{n}, f = (f_{1}, ..., f_{n}): ^{(ω)}ℍ^{n} → ^{(ω)}ℍ^{n}, c ∈ [a, b[A, if we choose ↷B γ(x) = γ(↷D x), (so especially for A = ^{ω}ℝ, B maximal in ^{ω}ℂ^{2} and D maximal in ^{ω}ℝ^{2}) is exactly B-differentiable and it applies for all x ∈ [a, b[A und z = γ(x)

dB(F(z)) = dD(F ∘ γ)(x) = f(γ(x))γ′_{↷}D(x)dDx = f(z)dBz.

Proof:

Second fundamental theorem of exact calculus for line integrals: If F is, instead of f as above, for t ∈ [a, b[A right-sided exactly successor-differentiable and its exact B-successor-derivative F′_{r}B is there exactly B-line integrable, then it applies, if we choose ↷B γ(t) = γ(↷D t), (so especially for A = ^{ω}ℝ, B maximal in ^{ω}ℂ^{2} and D maximal in ^{ω}ℝ^{2}) and γ: [a, b[A → ^{(ω)}ℍ^{n}:

Proof: Because of

(F(↷B γ(t)) - F(γ(t)) (↷B γ(t) - γ(t))/(↷D t - t) = (F(γ(↷D t) - F(γ(t)) γ′_{↷}D(t) = F′_{↷}B(γ(t))γ′_{↷}D(t) (↷B γ(t) - γ(t)) = (F ∘ γ)′_{↷}D(t) γ′_{↷}D(t)(↷D t - t)

it applies

Corollary: It applies for a closed path γ: [a, b[A → ^{(ω)}ℍ^{n} and f = (f_{1}, ..., f_{n}): ^{(ω)}ℍ^{n} → ^{(ω)}ℍ^{n}

if f has an antiderivative F on γ and ↷B γ(t) = γ(↷D t) for t ∈ [a, b[A applies.

Remark: In both fundamental theorems, the conventionally real case results analogously as above. For v, w ∈ [a, b[A, v ≠ w and γ(v) = γ(w) ↷B γ(v) ≠ ↷B γ(w) is permitted. Notice that continuity is not presupposed for integral and derivative. Actual integration (as inversion of the derivative) only makes sense for continuous functions, if it is to go beyond mere summation. However, if function values can be combined into a finite number of continuous functions, for which each of them the antiderivative can be specified in finite time, also the integral for discontinuous functions can be calculated in this way, possibly with the appropriate aid of the Euler-Maclaurin sum formula and further simplification techniques.

Remark: In the same way, Stokes' theorem (see below) can be proven, since the integrals vanish on the inner paths. Over the more or less elements is integrated, the greater the value of the integral can differ, even within the same interval limits. If we use the alternative exact derivation, then the formulas change accordingly, and this applies the less, the more continuous the occurring functions are. Here and in general appropriate rounding rules can be helpful.

Definition: The tightened *(right-sided) exact B-integral according to the trapezoid rule* is defined by

The tightened *(right-sided) exact B-integral according to the midpoint rule* is - the existence of (z + ↷B z)/2 presupposed - defined by

Remark: Since these tightened exact B-integrals are obviously independent from the direction of traversing, theorems are justified by their (implicit) use, for which results of integration for opposite traversing shall just become zero as e.g. Green's theorem (see below). In the first fundamental theorem, the derivation dB(F(z))/dBz is tightened to the arithmetical mean (f(z) + f(↷B z))/2 resp. to (f(z + ↷B z)/2), in the second one F(γ(b)) - F(γ(a)) to (F(γ(b)) + F(↶B γ(b)))/2 - (F(γ(a)) + F(↷B γ(a)))/2 resp. to F((γ(b) + ↶B γ(b))/2) - F((γ(a) + ↷B γ(a))/2), where in the sufficiently α-continuous case of f resp. of F at the boundary nearly the original results emerge. Since the exact B-integral is defined according to a rectangle rule, corresponding error estimations arise concerning the exact integration (see references to numerical analysis).

Leibniz' differentiation rule: For f: ^{(ω)}ℍ^{n+1} → ^{(ω)}ℍ, a, b: ^{(ω)}ℍ^{n} → ^{(ω)}ℍ, ↷B x := (s, x_{2}, …, x_{n})^{T}, s ∈ ^{(ω)}ℍ \ {x_{1}} we have, if we choose ↷D a(x) = a(↷B x) and ↷D b(x) = b(↷B x),

Proof:

Annotation: Here it is integrated in the complex over a path whose starting and end point form the limits of the integral. If ↷D a(x) ≠ a(↷B x), so the last addend is to multiply by (↷D a(x) - a(x))/(a(↷B x) - a(x)) and if ↷D b(x) ≠ b(↷B x), so the second last addend is to multiply by(↷D b(x) - b(x))/(b(↷B x) - b(x)).

Definition: For a close path γ: [a, b[A → ^{(ω)}ℂ and z ∈ ^{(ω)}ℂ

is called the *winding number* or *index* ind_{γ}(z).

Integral formula: With f: A → ^{(ω)}ℂ and γ([a, b[) ⊆ A ⊆ ^{(ω)}ℂ, it is

iff with g(ζ) = (f(ζ) - f(z))/(ζ - z)

applies, thus especially if g has on γ an antiderivative.

Proof: The assertion follows directly from the corollary to the second fundamental theorem.⃞

Remark: The winding number is 0, if z is not revolved (antiderivative ln(ζ - z)). With n ∈ ℕ, it is n (-n) if it is revolved n times positively (negatively). This can be easily seen from the following figure with the parametrisation ζ = z + r e^{it} of the circle line λ with the length 2πr for t ∈ [0, 2π[ and r ∈ ℝ_{>0}:

Fig. 1

Example: The integration on the boundary ∂đ of the unit disk đ resp. its translation by 2 yields over 1/z the values 2πi resp. 0 (antiderivative ln(z + 2) on ∂đ + 2), and over |z|^{2} 0 (antiderivative z on ∂đ) resp. 4πi.

Remark: The example shows that the existence of a antiderivative of the integrand of a path integral and its value generally depend only on the integrand and path, including its orientation, not on the property of the underlying set, the interior and exterior of a path, the holomorphy or the homology (to zero) as in the conventional complex analysis.

Equation of the mean value: If we have γ([0, 2π[) = ∂Br(c) with c ∈ ^{(ω)}ℂ and r ∈ ^{(ω)}ℝ_{>0}, we obtain, provided f: B_{r}(c) → ^{(ω)}ℂ satisfies the condition of the integral formula,

Proof: The substitution z = c + e^{iφ} in the integral formula yields directly the assertion.⃞

Remark: With the standard estimate, we receive thereof the

inequality of the mean value: |f(c)| ≤ |f|_{γ}.

Definition: The coefficient a_{-1} of the function f: A → ^{(ω)}ℂ with A ⊆ ^{(ω)}ℂ and

and n ∈ ℕ, a_{i}, c, a_{ij}, c_{i} ∈ ^{(ω)}ℂ as well as pairwisely different c_{i} ≠ c is called the *residue* res_{c} f.

Residue theorem: If f: A → ^{(ω)}ℂ with γ([a, b[) ⊆ A ⊆ ^{(ω)}ℂ is represented by

with n ∈ ℕ, a_{ij}, c_{i} ∈ ^{(ω)}ℂ and c_{i} pairwisely different, it applies

for a closed path γ: [a, b[ → ^{(ω)}ℂ.

Proof: For all i ∈ ^{ω}ℕ_{≤n} and all j ∈ ℤ \ {-1}, we have

and

⃞

Intermediate value theorem: Let be f: [a, b] → ^{(ω)}ℝ α-continuous in [a, b]. Then f(x) maps to any value between min f (x) and max f(x), for x ∈ [a, b], with an accuracy < α. If f is continuous in ^{ω}ℝ, it maps to any value of ^{κ}ℝ between min f(x) and max f(x).

Proof: Between min f(x) and max f(x), there exists an unbroken chain of overlapping α-environments, each with f(x) as the centre, since otherwise a contradiction to the α-continuity of f would emerge. The second part of the assertion follows from the fact that a deviation f(↷x) - f(x)| < 1/κ resp. |f(x) - f(↶x)| < 1/κ in ^{κ}ℝ, falls below the resolution maximally permitted.⃞

Extremum criterion: Iff f has, as above, in the point x_{0} a left-sided exact derivative > 0 and a right-sided exact derivative < 0, f has there a local maximum. Iff f has, as above, in the point x_{0} a left-sided exact derivative < 0 and a right-sided exact derivative > 0, f has there a local minimum.

Proof: Clear from the definitions.⃞

Definition: We define the derivative of a function f: A → ^{(ω)}ℝ with A ⊆ ^{(ω)}ℝ as 0, iff 0 lies in the interval with the boundaries of the left- and right-sided exact derivative. The corresponding argument of the function is then called *stationary point*.

Remark: More precisely, we can define the (left-) right-stationary point, for which the (left-) right-sided exact derivative is 0.

Example: The (2d0)-continuous function f: ^{(ω)}ℝ → {0, d0} with

consists only of the local minima 0 and the local maxima d0, and has only the (left- or right-sided) exact derivatives ±1.

Rolle's theorem: If f: [a, b] → ^{(ω)}ℝ with a, b ∈ ^{(ω)}ℝ and a < b is not strictly monotone, then there is an y ∈ [a, b] with f′B(y) = 0.

Proof: Since f has in [a, b] at least an extremum or a stationary point, the assertion follows directly.⃞

First mean value theorem of differential calculus: Let be f: [a, b] → ^{(ω)}ℝ with a, b ∈ ^{(ω)}ℝ and a < b B-differentiable there. Then there is a point y ∈ [a, b] with

Proof: Since

is not strictly monotone because of h(a) = h(b), the assertion follows for h′B(y) = 0 with a y ∈ [a, b] from the preceding theorem by the exact B-derivative of h(x).⃞

Second mean value theorem of differential calculus: Let be f, g: [a, b] → ^{(ω)}ℝ with a, b ∈ ^{(ω)}ℝ and a < b B-differentiable there with g′ ≠ 0 (and thus g(a) ≠ g(b) after Rolle's theorem). Then there is a point y ∈ [a, b] with

Proof: We set

and conclude the assertion analogously as before.⃞

Remark: If we drop g′ ≠ 0 only f′B(y)(g(a) - g(b)) = g′B(y)((f(a) - f(b))) applies.

Definition: Let be f: A → ^{(ω)}ℍ with A ⊆ ^{(ω)}ℍ. Then

is called the *second derivative* towards ↷B z of f in z ∈ A.

Higher (partial) derivatives are defined analogously. The number j ∈ ℕ of executed partial derivatives is specified as exponent behind ∂; the variables, after which is differentiated, follow each other in the denominator, each one preceded by ∂. Hereby, the same variables are provided with an exponent, according to the number of their occurrence. Taylor series make only sense for |ℕ*|-fold α-continuously differentiable functions, because of their approximating and convergence behaviour.

Point-of-inflection criterion: Iff f has, as above, in the point x_{0} a left-sided exact second derivative > 0 and a right-sided exact derivative < 0, f has there a local point of inflection from left to right. Iff f has, as above, in the point x_{0} a left-sided exact derivative < 0 and a right-sided exact second derivative > 0, f has there a local point of inflection from right to left.

Proof: Equally clear from the definitions.⃞

Exchange theorem: The result of multiple partial derivatives of a function f: A → ^{(ω)}ℍ is independent from the order, whilst variables are replaced by values or limit values are considered only finally, if necessary (*latest-insert-principle*).

Proof: The derivative is uniquely determined: This is clear until the second derivative, for higher ones the assertion follows by (transfinite) mathematical induction.⃞

Example: Let be f: ^{ω}ℝ^{2} → ^{ω}ℝ given by f(0, 0) = 0 and f(x, y) = xy^{3}/(x^{2} + y^{2}) else. Then it applies:

with value ½ for (0, 0), although in

we have y for x = 0 on the left and 0 for y = 0 on the right, i.e. then, the partial derivative towards y yields on the left 1 ≠ 0, which is the partial derivative towards x on the right.

Theorem: Iff F: A → ^{(ω)}ℂ, with dividing in the real and imaginary part, F(z) := U(z) + i V(z) := f(x, y) := u(x, y) + i v(x, y), infinitesimal h = |dBx| = |dBy|, h-homogeneous A ⊆ ^{(ω)}ℂ, the neighbourhood relation B ⊆ A^{2} for all z = x + i y ∈ A, is holomorphic and

is infinitesimal, we receive the *Cauchy-Riemann partial differential equations*

if B is true for both ↷ and for ↶.

Proof: Since with

and dBz = dBx + i dBy all derivatives defined in A are given and since we have

and, with analogous formulas for v as also for ↶

with the last addend to be neglected exactly under the condition stated, we get the assertion.⃞

Remark: The last addend can especially be neglected, if f is continuous. From the equations for F′B(z), we obtain then the necessary and sufficient condition for the holomorphy of F

Definition: A (infinitely) real-valued function with arguments ∈ ^{(ω)}ℍ^{n} is called *convex (concave)* if its complete function values of the arguments between two different of its arguments, if any, in each case are not above (not below) of each connecting line segment of the function values of these arguments. This applies *strictly* if we can replace not above (not below) by below (above).

Definition: A sequence of adjacent h-cubes, which at least embrace a quadratic h-area not belonging to them, is called *closed chain* of h-cubes. An h-homogeneous set A ⊆ ^{(ω)}ℍ^{n} is called *simply connected*, if it contains no closed chain of h-cubes. It is called *connected*, if every h-cube adjacent to their other ones is connected with each of their other ones. A simply h-homogeneous set A ⊆ ^{(ω)}ℍ^{n} is called *h-convex*, if the points that are on the connecting distance between two arbitrary vertices of two different h-cubes are also vertices of h-cubes from A there to be expected. It is called *star h-domain*, if the points that are on the connecting distance from a (central) vertex of a h-cube from A and an arbitrary vertex of an arbitrary further h-cube from A are also vertices from h-cubes there to be expected. Let a single h-cube both h-convex and star h-domain.

Definition: We declare for the counter-directional integration over identical paths in positive and negative orientation as *counter-directional rule* for integrals that the function value for the successor of the argument must be selected in negative orientation so that the value of the integral over both directions is just 0.

Note: The counter-directional rule is necessary because the corresponding integrals may have there, where we would expect the value 0, another (significant) value.

Green's theorem: It applies for the neighbourhood relations B ⊆ A^{2} with simply connected h-set A ⊆ ^{(ω)}ℝ^{2}, infinitesimal h = |dBx|= |dBy| = |↷B γ(t) - γ(t)| = O(ω^{-m}), sufficiently great m ∈ ℕ*, (x, y) ∈ A, a closed path γ: [a, b[→ ∂A, traversed in both directions, if we choose ↷B γ(t) = γ( ↷D t), t ∈ [a, b[, D ⊆ [a, b]^{2} and sufficiently α-continuous functions u, v: → ℝ, with (not necessarily continuous) partial derivatives ∂Bu/∂Bx, ∂Bu/∂By, ∂Bv/∂Bx and ∂Bv/∂By

Proof: W.l.o.g. the line of argument follows only for A := {(x, y) : c ≤ x ≤ d, f(x) ≤ y ≤ g(x)}, c, d ∈ ^{(ω)}ℝ, f, g : ∂A → ^{(ω)}ℝ, since it runs for the equivalent, each turned by 90°, analogously and each simply connected h-set is a union of such sets. Only

is shown, since the missing relation results analogously. Since the parts of γ with dBx = 0 do not contribute to the line integral, we have with negligible t := h(u(d, g(d)) – u(c, g(c)))

⃞

Remark: The choice of m depends on the number of sets needed, having the types mentioned in the proof, whose union results in the simply connected h-set.

Goursat's integral lemma: If f is holomorphic in a triangle Δ ⊆ ^{(ω)}ℂ and if f has there no antiderivative, it applies

Refutation: It is apparently irrelevant in which direction we traverse ∂Δ. If Δ is completely triangulated, then for every minimal subtriangle Δ_{s} ⊆ Δ we must have w.l.o.g. either

or

with the vertices z_{1}, z_{2} and z_{3} of Δ_{s}. Due to the holomorphy or circular permutation, this can only occur for f(z_{1}) = f(z_{2}) = f(z_{3}). If we include all adjacent subtriangles in Δ, f must therefore be constant, in contradiction to the assumption. For, since the term in the great bracket is translationally invariant, we could otherwise set w.l.o.g. z_{3} := 0, and this term would only be 0 if we have z_{1} = z_{2}(1 ± i√3)/2 with |z_{1}| = |z_{2}| = |z_{1} – z_{2}|. However, since each horizontal and vertical straight line is homogeneous in ^{(ω)}ℂ, this cannot take place, because the corresponding subtriangle would be equilateral, and not isosceles and right-angled. Thus |I_{s}| is in both cases at least |f′(z_{2}) O(d0^{2})|, if we choose w.l.o.g. the vertices 0, |d0| and i|d0|. If L indicates the perimeter of a triangle, we have on the one hand |I| ≤ 4^{m} |I_{s}| with infinite natural m and on the other hand 2^{m} = L(∂Δ)/ |O(d0^{2})| because of L(∂Δ) = 2^{m} L(∂Δ_{s}) and L(∂Δ_{s}) = |O(d0^{2})|. Therefore |I| ≤ |f′(z_{2}) L(∂Δ)^{2}/O(d0^{2})| applies, so that the desired estimate |I| ≤ |O(dBζ)| fails, if say |f′(z_{2}) L(∂Δ)^{2}| is greater than |O(d0^{2})|.⃞

Cauchy's integral theorem: It applies for the neighbourhood relations B ⊆ A^{2} and D ⊆ [a, b] with a simply connected h-set A ⊆ ^{ω}ℂ, infinitesimal h as well as a holomorphic function f: A → ^{ω}ℂ and a closed path γ: [a, b[→ ∂A, if we choose ↷B γ(t) = γ(↷D t) with t ∈ [a, b[,

Proof: Because of the Cauchy-Riemann partial differential equations, we have with x := Re z, y := Im z, u := Re f, v := Im f and A^{‒} := {z ∈ A : z + h + ih ∈ A}

Fundamental theorem of algebra: For every non-constant polynomial P ∈ ^{(ω)}ℂ, it exists a z ∈ ^{(ω)}ℂ with P(z) = 0.

Indirect proof: We can achieve 1/P(0) ≠ O(d0) by affine-linear variable substitutions. We assume P(z) ≠ 0 for all z ∈ ^{(ω)}ℂ. We have for the holomorphic f(z) := 1/P(z) f(1/d0) = O(d0) and, because of the inequality of the mean value, |f(0)| ≤ |f|_{γ} with γ = ∂B_{r}(0) and arbitrary r ∈ ^{(ω)}ℝ_{>0}, thus f(0) = O(d0) in contradiction to the presupposition.⃞

Stokes' theorem: If ^ stands above a term to omit and

is an alternating differential form of degree n - 1 on an axially parallel cuboid C = [a_{1}, b_{1}]×…×[a_{n}, b_{n}] ⊆ ^{(ω)}ℝ^{n} with sufficiently α-continuous functions f_{k}: C → ^{(ω)}ℝ, where

with the faces F_{a,k} = [a_{1}, b_{1}]×… ×{a_{k}}×… ×[a_{n}, b_{n}] and F_{b,k} = [a_{1}, b_{1}]×… ×{b_{k}}×… ×[a_{n}, b_{n}] of C, then applies

Proof: Because of

and

the assertion follows with the second fundamental theorem of exact calculus and Fubini's theorem.⃞

Remark: Since n-dimensional manifolds can be assembled, as would seem natural, from the cuboids above, the theorem applies also for them as for more general differential forms. Integrals over higher dimensional volumes can be dissolved into such over lower dimensional volumes, if the same volumes lying opposite to each other are oriented in opposite direction for cubes with edge length d0.

Let be all conventionally complex functions f: A → ^{(ω)}ℂ for A ⊆ ^{(ω)}ℂ implicitly so defined that we have max |f(z)| ≤ r := max ^{(ω)}ℝ and f(z) = r f(z)/|f(z)| for actually |f(z)| > r and also always for f = id (i.e. we note differently ^{(ω)}ℂ for B_{≤r } (0)). The entire functions f(z) = z/ω and g(z) = Σa_{k} z^{k} with k ∈ ℕ and a_{k} = 1/ω^{k+1} disprove Liouville's and Picard's little theorem.

Evidence: Since we have |f(z)| ≤ 1 and |g(z)| < 1, the assertion follows directly.⃞

Remark: Choosing sufficiently small (transcendental) constants in the generalisation of Liouville's theorem disproves it too. Both theorems cannot be remedied by restricting them, since the holomorphy of a function h on ℂ compels that the Laurent polynomial (the Laurent series) of h must have coefficients a_{k} with |a_{k}| < O(ω^{-|k|}) (O(ω^{-|k|-1})) and (infinite) integer k (to converge), if it is not already constant. Thus, a limitation to coefficients ≥ 1/ω is pointless.

The function f yields the biholomorphic, bijective mapping of the circular defined ^{κ}ℂ onto the complex unit disk đ_{d}, very condensed from the complex unit disk đ, with |đ_{d}| = |^{κ}ℂ| ≫ |đ|. Therewith, the Riemann mapping theorem is also valid for ℂ. The complete ^{κ}ℂ could not be mapped this way, of course. The function 1/f disproves Picard's great theorem, if ^{ω}ℂ is assumed arbitrarily dense.

Definition: A point z_{0} ∈ M ⊆ ^{(ω)}ℂ^{n} resp. concerning a sequence (a_{k}) with a_{k} ∈ ^{(ω)}ℂ^{n} and (infinite) natural k is called *actual α-accumulation point* of M resp. of the sequence, if there are in the ball B_{α}(z_{0}) ⊆ ^{(ω)}ℂ^{n} around z_{0}, with the infinitesimal radius α infinitely many points of M resp. infinitely many pairwise different sequence members. If the asserted applies for α = 1/ω, the α-accumulation point is simply called accumulation point.

Let be p(z) = π(z - c_{k}) with k ∈ ^{ω}ℕ for z ∈ ^{ω}ℂ an infinite product with pairwise distinct zeros c_{k} ∈ B_{1/⌊ω⌋}(0) ⊂ ℂ (disks around 0 with radius 1/⌊ω⌋), which are chosen so that |f(c_{k})| < 1/⌊ω⌋ applies, for a in a domain G ⊆ ^{ω}ℂ holomorphic function f with f(0) = 0. G contains B_{1/⌊ω⌋}(0) completely, which is always obtainable by coordinate transformation, while G is "big" enough.

Then, for the also there holomorphic function g(z) := f(z) + p(z), the coincidence set {ζ ∈ G : f(ζ) = g(ζ)} has an accumulation point at 0, and we have f ≠ g, in contradiction to the statement of the identity theorem. Examples of f are all in B_{1/⌊ω⌋}(0) bounded and at the same time in G holomorphic functions with zero 0. Since p(z) can attain any conventionally complex value, the deviation from f and g is not negligible.

Also in contradiction to the identity theorem is the fact that, at a point z_{0} ∈ G all the derivatives d^{(n)}(z_{0}) = h^{(n)}(z_{0}) of two functions d and h can coincide for all n, but d and h can also be significantly different further away, beyond this local fact, without losing their holomorphy, since not every holomorphic function, due to the approximate character (of differentiation) resp. of the calculation with Landau symbols, can be (uniquely) expanded into a Taylor series (cf. Transcendental Numbers).

If we choose k ∈ ℕ factors in π(z - c_{k}), we obtain entire functions with infinite naturally many zeros. The zero set needs not to be discrete. Thus the set of all functions holomorphic in a domain G need not to be free from zero divisors. The functions disprove, like polynomials with at least n > 2 pairwise distinct zeros, again the little theorem of Picard, since they miss at least n - 1 values in ^{ω}ℂ.

Remark: We can approximate ζ(2n+1) for n ∈ ^{ω}ℕ* by

Here, we can decompound the addends of the first sum into a sum of partial fractions with the n-th roots of unity as zeros and continue splitting off until there is a sufficient quality of approximation, for the last sum is just ζ(4n+2). In doing so,

is characterised with z ∈ ^{ω}ℂ \ ^{ω}ℕ* by z and the difference to ln ⌈ω⌉. We can expand this splitting off method as follows (generalisation to infinite numbers possible):

Proposition: A series of similarly constructed members of fractions of polynomials in k ∈ ^{ω}ℕ*, with complex rational coefficients, can be represented, except a predetermined conventionally real precision, as the sum of ζ(2n) values, with n ∈ ^{ω}ℕ*, and F′(z) values, with z ∈ ^{ω}ℂ, multiplied by conventionally complex constants and a conventionally complex constant.

Proof: Successive partial fraction decomposition and estimate of the fast converging residual series.⃞

Remark: We can well compute the values of F′(z) with the Euler-Maclaurin sum formula or, in the real, with the Euler transformation, using the harmonic series. The uniform method for this series enables convergence acceleration and avoids the more difficult integration or the derivatives of the polynomial quotients, if we would do this directly, and establishes a value-based classification of the series, additively growing according to increasing precision, which can be exactly specified except the precision itself.

Examples: We obtain with the second Bernoulli numbers B_{k} and Faulhaber's formula as well as k ∈ ^{ω}ℕ, m, n ∈ ^{ω}ℕ* (generalisation to greater numbers possible):

We obtain from the identity

for real or complex x with (-x)^{k} := (-1)^{k} x^{k} by differentiation

if the modulus of x has another order than that of dx or 1/dx.

This formula can still be simplified to -1/(1+x)^{2} for sufficiently, but not too small x and sufficiently, but not too great (infinite) n, and remains also valid for not too great x ≥ 1. Further exact formulas for P_{j+1}(x) as examples of also divergent series, which were so far not always correctly computed, result from successively multiplying P_{j}(x) := P_{0}^{(j)}(x) by x for j ∈ ^{ω}ℕ* and subsequent differentiation.

P_{0}(z) applies for all complex z, what follows for the problematic case z = -1 from L'Hôpital's rule. If we integrate however P_{0}(-x) from 0 until 1 and set n := |ℕ*|, so we obtain an integral expression for ln |ℕ*| + γ with Euler's constant γ. If we substitute in it y := 1 - x, so we obtain for it via the binomial series a series with nearly only infinite coefficients; if we represent moreover ln |ℕ*| as series, we do so even for γ.

Remark: If we simplify the numerator of P_{0}(z) unduly to 1, we may obtain wrong results, especially if we have |z| ≥ 1.

Example: P_{0}(-e^{iπ}) equals 0 for odd n and 1 for even n, but does not equal ½.

Theorem: For

with x ∈ [1/κ - 1, 1 - 1/κ], there is no further expansion of a power series from that the ζ(2n+1) could be determined by coefficient comparison.

Proof: If we assume the functional equation f(x + 1) = f(x) - 1/(x + 1) + 1/(x + ⌈ω⌉), so even

is there uniquely determined, except a constant addend, where we have only |x + k| ≥ 1/κ for all k ∈ ℕ*. If we add an arbitrary power series P(x) to f(x), so the assertion follows analogously.⃞

© 22.02.2015 by Boris Haase

• disclaimer • mail@boris-haase.de • pdf-version • bibliography • subjects • definitions • statistics • php-code • rss-feed • top