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#62: Improvement Nonstandardanalysis on 22.02.2015

Preliminary remark: In the following, the definitions from the set theory are applied and let be at most m, n ∈ ωℕ*. The integration and differentiation on arbitrary always non-empty subsets A of the sets (ω)n or (ω)n with arbitrary n are studied, which we combine into the sets (ω)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)>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)>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 κn, with n ∈ ωℕ*, 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 ⊆ A2. The set ÑB(z0) of all neighbours of z0 ∈ A (with respect to B) in A is called neighbourhood of z0. The function γ: C → A ⊆ ℂn 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 (z0, ↷z0) or (↶z0, z0), whereby ↷ and ↶ are pronounced "suc" and "pre". This applies analogously for the neighbourhood relation D ⊆ C2.

Definition: Let be z0 ∈ A ⊆ ℍn and f: A → (κ)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 z0 in direction ↷B z0 if for infinitesimal α ∈ (ω)>0 applies: :

||f(↷B z0) - f(z0)|| < 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 z0 and ↷B z0 ∈ ÑB(z0) 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)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)m,

d↷B zf(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 zf(z) by d((↷B)z1, ... , (↷B)zn)f(z). If f is the identity, that is f(z) = z, we can write d↷B zBz instead of d↷B zf(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 fk(↷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 fk(↷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 ωℕ is not of this type, since we have s: ωℕ → ωℕ* ∪ {|ωℕ|}.

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

formula_094

Obviously, g is completely defined and continuous by this specification. Let the function Φ: I = [0, 1] → ℝ2 be defined by

formula_095"

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 ⊆ ωℝ → ωℝ is called jump discontinuity with jump to the right (left) of s:= |f(↷x) – f(x)| upward (downward) or vice versa, if s > 1/|ωℕ*| applies.

Theorem: A monotone function f: [a, b] → ωℝ has maximally |ωℕ||ωℤ*| - 1 jump discontinuities.

Proof: Between -|ωℕ*| and |ωℕ*|, maximally |ωℕ*||ωℤ*| jump discontinuities with jump 1/|ωℕ*| are possible, beyond that, taken together, maximally |ωℤ*| - 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)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} = Br(a)° is correspondingly called inner ball.

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

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

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

formula_167

It is moreover strictly monotone, i.e. for h-homogeneous sets A1, A2(ω)n with A1 ⊂ A2 applies µh(A1) < µh(A2). If h is not equal for all considered sets Ak, 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 µd0(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 µd0(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 zk of F: A → (ω)ℍ in z = (z1, ..., zn) ∈ A ⊆ (ω)n with k ∈ [1, n]ℕ is defined as

formula_117

If we have, with the notations above,

formula_118

for a function f = (f1, ..., fn): A → (ω)n with z ∈ A ⊆ (ω)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 Fl(x) and Fr(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′rB(w), and for ↷B w < w we do so of the left-sided exact derivative F′lB(w). If the derivatives match towards all neighbours, we speak correspondingly of the exact derivative F′B(z) (for A = κℂ 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 ⊆ A2, f: A → C ⊆ (ω)ℝ, D ⊆ C2, g: C → (ω)ℝ we have, if we choose f(↷B x) = ↷D f(x),

g′rB(f(x)) = g′rD(f(x)) f′rB(x).

Proof:

formula_177

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

formula_159

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-1r(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

formula_077

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

formula_199

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′bB(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)n as well as where dBz resp. ↷B z and the right-hand side exist in ℍn,

formula_045

is called the exact B-integral of a vector field f = (f1, ..., fn): A → (ω)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)n, A ⊆ ℝ and f = (f1, ..., fn): (ω)n(ω)n

formula_120

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 ω2 and D maximal in ω2), is called the exact B-line integral of a vector field f along the path γ, if the right-hand integral exists in ℍn. 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

formula_141

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 = A1× ... ×An(ω)n with for each zk ∈ Ak in each case uniquely determined neighbour ↷Bk zk of the neighbourhood relations Bk ⊆ Ak×(ω)ℍ, for all k ∈ ωℕ*≤n and B = B1× ... ×Bn

formula_121

is called the exact B-volume integral over a then B-volume integrable function f = (f1, ... , fm): A → (ω)m, if the right-hand side exists in ωn. Improper exact B-volume integrals are defined analogously to the exact B-integrals.

Remark: Obviously, we have

formula_122

if dBxk = h is fulfilled for all x of a h-homogeneous set A ⊆ (ω)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]ωℤ a non-empty h-homogeneous subset of [a, b[ωℝ with B = ]a, b]h[1, n]ωℤ. For h = 1/κ and κ = -a = b – h, [a, b[h[1, n]ωℤ is comparable with κℝ. Let be Tr furthermore a right-sided B-antiderivative of a Taylor series t, not necessarily convergent in [a, b[h[1, n]ωℤ, 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]ωℤ, but it applies exactly for all h

formula_052

and

formula_053

Example: The middle thirds Cantor set C has the relative measure µd0(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

formula_169

Example: For the classes a + ωℚ with a ∈ ωℝ the equivalence relation x ~ y ⇔ x - y ∈ ωℚ with x and y ∈ ωℝ their representatives can be specified through the set R = [0, 1/(⌊ω⌋)[ with the measure µd0(R) = 1/⌊ω⌋. Let the function r: ωℝ → {0, 1} be defined by r(x) = 1 for x ∈ ωℤ + R and r(x) = 0 for x ∈ ωℝ \ (ωℤ + R). Then it applies

formula_170

Example (cf. set theory): Let A1 = [0, 1[ ∩ ωA and the function q: A1 → {0, 1} be defined by q(x) = 1 for x ∈ A1 \ ωℚ and q(x) = 0 for x ∈ A1ωℚ. The exact integral over q(x)dx has then in A1 the transcendental value

formula_171

Remark: The sets C, R and ωℚ 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 (ak) with members ak is a map from (ω)ℤ to (ω)m: k ↦ ak. A series is a sequence (sk) with m ∈ (ω)ℤ and the partial sums

formula_166

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 A1, A2(ω)ℍ, it applies with f: A1×A2(ω)

formula_143

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

Example: Because of

formula_144

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

formula_145

as the case may be, and we do not have

formula_146

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 (sk) for infinite (infinite) natural k and j is finite, iff it can be represented as

formula_058

with finite ||aj - bj|| forming a monotonically nonincreasing sequence for aj, bj(ω)m.

Proof: The assertion follows directly from the finiteness of ||a1 - b1|| 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

formula_172

Remark: More interesting examples are those whose nonincreasing monotony of |aj - bj| cannot be as easily proven as e.g. for the divergent series

formula_173

where cj increases monotonically, but cj+1 - cj 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 anxn is conventionally real, if the radius of convergence of the latter is ≤ s/lim sup |an|1/n.

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

formula_061

for k ∈ ℕ* and akκ>0 is finite, if with finite

formula_062

eS is maximally finite.

Proof: With the exponential series, the assertion follows directly from S < P < eS.⃞

Remark: Products with akκℂ 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 (ak) with k ∈ (ω)ℕ*, ak(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 ak with k ≥ q and not too small difference max k - q |ak - a| < α applies. The set α-A of all such a is called α-set of limit values of (ak), 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)>0 - in particular by the formulation for all conventionally real ε > 0 - that a real number ε/r with a real r ∈ (ω)>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 ∈ ωℕ* and x ∈ [0, 1]):

1. The sequence fn(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 fn(x) = x - xn/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 fn(x) = 1 - xn-1 is discontinuous in the point x = 1.

3. The sequence fn(x) = (n2/2 - |n3(x - 1/(2n))|)(1 - sgn(x - 1/n)) (or expressed with continuously differentiable functions

formula_104

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

formula_105

and

formula_182

instead of

formula_107

because of allegedly f(x) = 0.

4. The sequence fn(x) = (n/2 - |n2(x - 1/(2n))|)(1 - sgn(x - 1/n)) (or expressed with continuously differentiable functions

formula_108

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

formula_109

instead of

formula_110

because of allegedly f(x) = 0.

5. The sequence fn(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 (fj) a sequence of integrable functions with (infinite) natural j and fj: A → (ω)ℝ, which α1-converge to the integrable function f: A → (ω)ℝ. Then applies for:

formula_174

formula_125

Proof:

formula_126.⃞

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 α1μd0(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

formula_067

with γ: [c, x[A → (ω)n, f = (f1, ..., fn): (ω)n(ω)n, c ∈ [a, b[A, if we choose ↷B γ(x) = γ(↷D x), (so especially for A = ωℝ, B maximal in ω2 and D maximal in ω2) 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:

formula_178

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′rB is there exactly B-line integrable, then it applies, if we choose ↷B γ(t) = γ(↷D t), (so especially for A = ωℝ, B maximal in ω2 and D maximal in ω2) and γ: [a, b[A → (ω)n:

formula_128

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

formula_179

Corollary: It applies for a closed path γ: [a, b[A → (ω)n and f = (f1, ..., fn): (ω)n(ω)n

formula_082

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

formula_161

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

formula_162

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)n+1(ω)ℍ, a, b: (ω)n(ω)ℍ, ↷B x := (s, x2, …, xn)T, s ∈ (ω)ℍ \ {x1} we have, if we choose ↷D a(x) = a(↷B x) and ↷D b(x) = b(↷B x),

formula_152

Proof:

formula_180

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)

formula_083

is called the winding number or index indγ(z).

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

formula_084

iff with g(ζ) = (f(ζ) - f(z))/(ζ - z)

formula_087

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 eit of the circle line λ with the length 2πr for t ∈ [0, 2π[ and r ∈ ℝ>0:

integration paths
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)>0, we obtain, provided f: Br(c) → (ω)ℂ satisfies the condition of the integral formula,

formula_142

Proof: The substitution z = c + e 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

formula_088

and n ∈ ℕ, ai, c, aij, ci(ω)ℂ as well as pairwisely different ci ≠ c is called the residue resc f.

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

formula_089

with n ∈ ℕ, aij, ci(ω)ℂ and ci pairwisely different, it applies

formula_090

for a closed path γ: [a, b[ → (ω)ℂ.

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

formula_091

and

formula_092

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 ωℝ, it maps to any value of κℝ 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 κℝ, falls below the resolution maximally permitted.⃞

Extremum criterion: Iff f has, as above, in the point x0 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 x0 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

formula_154

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

formula_155

Proof: Since

formula_156

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

formula_157

Proof: We set

formula_158

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

formula_078

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 x0 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 x0 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: ω2ωℝ given by f(0, 0) = 0 and f(x, y) = xy3/(x2 + y2) else. Then it applies:

formula_080

with value ½ for (0, 0), although in

formula_081

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 ⊆ A2 for all z = x + i y ∈ A, is holomorphic and

formula_130

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

formula_131

if B is true for both ↷ and for ↶.

Proof: Since with

formula_132

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

formula_133

and, with analogous formulas for v as also for ↶

formula_134

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

formula_140

Definition: A (infinitely) real-valued function with arguments ∈ (ω)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)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)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 ⊆ A2 with simply connected h-set A ⊆ (ω)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

formula_135

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

formula_136

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)))

formula_137

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

formula_111

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

formula_112

or

formula_113

with the vertices z1, z2 and z3 of Δs. Due to the holomorphy or circular permutation, this can only occur for f(z1) = f(z2) = f(z3). 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. z3 := 0, and this term would only be 0 if we have z1 = z2(1 ± i√3)/2 with |z1| = |z2| = |z1 – z2|. 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 |Is| is in both cases at least |f′(z2) O(d02)|, 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| ≤ 4m |Is| with infinite natural m and on the other hand 2m = L(∂Δ)/ |O(d02)| because of L(∂Δ) = 2m L(∂Δs) and L(∂Δs) = |O(d02)|. Therefore |I| ≤ |f′(z2) L(∂Δ)2/O(d02)| applies, so that the desired estimate |I| ≤ |O(dBζ)| fails, if say |f′(z2) L(∂Δ)2| is greater than |O(d02)|.⃞

Cauchy's integral theorem: It applies for the neighbourhood relations B ⊆ A2 and D ⊆ [a, b] with a simply connected h-set A ⊆ ωℂ, infinitesimal h as well as a holomorphic function f: A → ωℂ and a closed path γ: [a, b[→ ∂A, if we choose ↷B γ(t) = γ(↷D t) with t ∈ [a, b[,

formula_138

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}

formula_181

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 = ∂Br(0) and arbitrary r ∈ (ω)>0, thus f(0) = O(d0) in contradiction to the presupposition.⃞

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

formula_147

is an alternating differential form of degree n - 1 on an axially parallel cuboid C = [a1, b1]×…×[an, bn] ⊆ (ω)n with sufficiently α-continuous functions fk: C → (ω)ℝ, where

formula_148

with the faces Fa,k = [a1, b1]×… ×{ak}×… ×[an, bn] and Fb,k = [a1, b1]×… ×{bk}×… ×[an, bn] of C, then applies

formula_149

Proof: Because of

formula_150

and

formula_151

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) = Σak zk with k ∈ ℕ and ak = 1/ωk+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 ak with |ak| < 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 κℂ onto the complex unit disk đd, very condensed from the complex unit disk đ, with |đd| = |κℂ| ≫ |đ|. Therewith, the Riemann mapping theorem is also valid for ℂ. The complete κℂ could not be mapped this way, of course. The function 1/f disproves Picard's great theorem, if ωℂ is assumed arbitrarily dense.

Definition: A point z0 ∈ M ⊆ (ω)n resp. concerning a sequence (ak) with ak(ω)n and (infinite) natural k is called actual α-accumulation point of M resp. of the sequence, if there are in the ball Bα(z0) ⊆ (ω)n around z0, 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 - ck) with k ∈ ωℕ for z ∈ ωℂ an infinite product with pairwise distinct zeros ck ∈ B1/⌊ω⌋(0) ⊂ ℂ (disks around 0 with radius 1/⌊ω⌋), which are chosen so that |f(ck)| < 1/⌊ω⌋ applies, for a in a domain G ⊆ ωℂ holomorphic function f with f(0) = 0. G contains B1/⌊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 B1/⌊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 z0 ∈ G all the derivatives d(n)(z0) = h(n)(z0) 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 - ck), 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 ωℂ.

Remark: We can approximate ζ(2n+1) for n ∈ ωℕ* by

formula_096

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,

formula_097

is characterised with z ∈ ωℂ \ ωℕ* 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 ∈ ωℕ*, with complex rational coefficients, can be represented, except a predetermined conventionally real precision, as the sum of ζ(2n) values, with n ∈ ωℕ*, and F′(z) values, with z ∈ ωℂ, 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 Bk and Faulhaber's formula as well as k ∈ ωℕ, m, n ∈ ωℕ* (generalisation to greater numbers possible):

formula_098

We obtain from the identity

formula_099

for real or complex x with (-x)k := (-1)k xk by differentiation

formula_100

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 Pj+1(x) as examples of also divergent series, which were so far not always correctly computed, result from successively multiplying Pj(x) := P0(j)(x) by x for j ∈ ωℕ* and subsequent differentiation.

P0(z) applies for all complex z, what follows for the problematic case z = -1 from L'Hôpital's rule. If we integrate however P0(-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 P0(z) unduly to 1, we may obtain wrong results, especially if we have |z| ≥ 1.

Example: P0(-e) equals 0 for odd n and 1 for even n, but does not equal ½.

Theorem: For

formula_175

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

formula_176

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


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