## #60: Improvement Nonstandard Analysis on 21.06.2014

Preliminary remark: In the following, the definitions from the set theory are applied and let be at most m, n ∈ ℕ*. The integration and differentiation on arbitrary always non-empty subsets A of the sets (~)n or (~)n with arbitrary n are studied, which we combine into the sets (~)n = (~)ℍ× ... ×(~)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 → (~)>0 with the vector space V over (~)ℍ is called norm, if we have for all x, y ∈ V and all λ ∈ (~)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 ∈ (~)>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 cn, with n ∈ 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.

Definition: Since d0 := min ||y – z|| applies for all pairwise distinct y, z ∈ ~n := (~ℝ + i~ℝ)n and x = m d0 for all x ∈ ~ℝ with m ∈ ~ℤ is called d0-homogeneous. A subset A ⊆ ~n with h = k d0 for a k ∈ ~ℤ is called h-homogeneous, if there is at least in each case a shortest sequence of points from ~n between all pairwise different points y, z ∈ A, for which all intermediate points and the terminal point z that can only be reached by successive addition from y, by ±h in the real components or ±ih in the complex components, also exist in A. The length of the sequence is l = hs with s ∈ ~ℕ as their number of executed steps. A point v = (v1, ... , vp) ∈ A is there called neighbour to a point w = (w1, ... , wp) ∈ A with vj, wj~ℝ or vj, wj ∈ i~ℝ, if |vj – wj| ≤ h for all j ∈ ℕ *≤p and for at least one j |vj – wj| = h applies.

Determination: In the nonstandard analysis 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 ⊆ ~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 (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 ⊆ ~n and f: A → (c)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 ∈ (~)>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/|ℕ*|, 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 = (~)2n applies.

Definition: For f: A → (~)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.

Proposition: If we assume, with z ∈ A ⊆ (~)n, B ⊆ A2 and infinitesimal α ∈ (~)>0, f: A → (c)ℍ αB-successor-continuous and g: A → (~)ℍ βB-successor-continuous, then we have there, if at all, f ± g at least (α + β)B-, fg at least (α max |g| + β max |f|)B- and f/g at least ((α max |g| + β max |f|) / min |g|2)B-successor-continuous. Each predecessor-continuity results analogously, like the assertions for multiple combinations of these operations and functions by successive inserting.

Proof: The first assertion follows from the triangle inequality. For the product and the quotient of f and g, the assertions follow without notating B from

f(↷z)g(↷z) - f(z)g(z) = (f(↷z) - f(z))g(↷z) + f(z)(g(↷z) - g(z))

and

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 ℕ is not of this type, since we have s: ℕ → ℕ* ∪ {|ℕ|}.

Example of a Peano curve (translated from Wolfgang Walter, Analysis 2, 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] → ℝ2 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 ⊆ ℝ → 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/|ℕ*| applies.

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

Proof: Between -|ℕ*| and |ℕ*|, maximally |ℕ*||ℤ*| jump discontinuities with jump 1/|ℕ*| are possible, beyond that, taken together, maximally |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 c 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 := (~)n : ||z - a|| ≤ r} is called ball with radius r ∈ (~)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 → ~≥0 with a h-homogeneous m-dimensional set (~)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 (~)ℂ = 2 applies. If A is not h-homogeneous, so we set h := min ggT (|yk – zk|/d0) over all k ∈ ~ℕ*≤m as well as y, z ∈ A with |yk – zk| > 0 and h-homogenise A by adding all required intermediate points to A.

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

It is moreover strictly monotone, i.e. for h-homogeneous sets A1, A2(~)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: Let be A the subset of [0, 1[ that 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 (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 → (~)ℍ in z = (z1, ..., zn) ∈ A ⊆ (~)n with k ∈ [1, n]ℕ is defined as

If we have, with the notations above,

for a function f = (f1, ..., fn): A → (~)n with z ∈ A ⊆ (~)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 (~)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) (trans-) real, left-sided and right-sided B-antiderivatives Fl(x) and Fr(x) with x ∈ (~)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 ∈ (~)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 = cℂ and n = 1, F is then conventionally regarded as holomorphic).

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

where the quotient is defined. This has the advantage that F′bB(w) can be regarded more as "tangent slope" in the point w, which can become rather zero for a local extremum, especially if F is αB-continuous in w. This is convenient, for example, if we want to characterise the exact values of ↷B w and ↶B w only as arbitrarily close to w, or want to round the exact derivatives suitably, in order to provide simple derivation rules, if necessary. The transfer to the conventionally complex is done analogously.

Definition: With z ∈ A ⊆ (~)n as well as where dBz resp. ↷B z and the right-hand side exist in ℍn,

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

with dDt > 0, ↷D t ∈ ]a, b]A and ↷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 ℂ2 and D maximal in ℝ2), 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

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 (c)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.

Definition: For z ∈ A = A1× ... ×An ⊆ ℍn with for each zk ∈ Ak in each case uniquely determined neighbour ↷Bk zk of the neighbourhood relations Bk ⊆ Ak×ℍ, for all k ∈ ℕ*≤n and B = B1× ... ×Bn

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

Remark: Obviously, we have

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

and

Example: The middle thirds Cantor set C has the relative measure µd0(C) = (⅔)|ℕ*|. Let the function c: [0, 1] → {0, (⅔)-|ℕ*|} be defined by c(x) = (⅔)-|ℕ*| for x ∈ C and c(x) = 0 for x ∈ [0, 1] \ C. Then it applies

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

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

Remark: The sets C, R and 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 (ak) with members ak is a map of [1, p]~ℕ to (~)m: k ↦ ak. A series is a sequence (sk) with the partial sums

The smallest index in sk for j can be defined differently (e.g. 0 or -k).

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(~)ℍ, it applies with f: A1×A2(~)

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

with finite ||aj - bj|| forming a monotonically nonincreasing sequence for aj, bj(~)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

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

where cj increases monotonically, but cj+1 - cj is monotonically nonincreasing.

Theorem: The number s := η(1 + ϑ/⌈ω⌉)/κ (t := 1 - 1/ω⌊ω⌋ + ϑ) with η := κ - 1 and each a ϑ ∈ ]0, 1[ is an upper bound, which makes, inserted as argument into the geometric series, the sum of the latter (conventionally) real.

Proof: Because of η = O(1), the assertions follow for ϑ = 0 resp. 1 with easy calculation from

and

Remark: With the notation of the preceding theorem 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 (t/lim sup |an|1/n).

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

for k ∈ ~ℕ* and akc>0 is finite, if with concretely specifiable

eS is maximally concretely specifiable.

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

Remark: Products with akcℂ 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 ∈ (~)ℕ*, ak(~)ℂ and α ∈ ]0, 1/κ] is called α-convergent to a ∈ (~)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/|ℕ|) 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. Harro Heuser, Lehrbuch der Analysis Teil 1, p. 144), 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 ∈ (~)>0 - in particular by the formulation for all conventionally real ε > 0 - that a real number ε/r with a real 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 trans-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 ∈ ℕ* and x ∈ [0, 1]):

1. The sequence fn(x) = sin(nx)/√n does not tend for n → |ℕ*| to f(x) = 0, but to f(x) = sin(|ℕ*|x)/√|ℕ*| with the (continuous) derivative f′(x) = cos(|ℕ*|x) √|ℕ*| instead of f′(x) = 0.

2. The sequence fn(x) = x - xn/n does not tend for n → |ℕ*| to f(x) = x, but to f(x) = x - x|ℕ*|/|ℕ*| with the (continuous) derivative f′(x) = 1 - x|ℕ*|-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

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

and

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

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

because of allegedly f(x) = 0.

5. The sequence fn(x) = nx(1 - x)n does not tend for n → |ℕ*| to f(x) = 0, but to the continuous function f(x) = |ℕ*|x(1 - x)|ℕ*| and takes for x = 1/|ℕ*| the value 1/e.

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

Theorem about commuting α-limit values for the integration: Let be A ⊆ (~)ℝ and (fj) a sequence of integrable functions with (trans-) natural j and fj: A → (~)ℝ, which α1-converge to the integrable function f: A → (~)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 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

with γ: [c, x[C → (~)n, C ⊆ ℝ, f = (f1, ..., fn): (~)n(~)n, c ∈ [a, b[C and ↷B γ(x) = γ(↷D x) (so especially for C = ℝ, B maximal in ℂ2 and D maximal in ℝ2) is exactly B-differentiable and it applies for all x ∈ [a, b[C 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[C right-sided exactly successor-differentiable and its exact B-successor-derivative F′rB is there exactly B-line integrable, then it applies for ↷B γ(t) = γ(↷D t) (so especially for C = ℝ, B maximal in ℂ2 and D maximal in ℝ2) and γ: [a, b[C → (~)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: For a closed path γ, we have with the conditions above that

whenever f has an antiderivative F on γ.

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 can be proven (see below), 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: For a close path γ: [a, b[A → (~)ℂ and z ∈ (~)

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

Integral formula: With f: A → (~)ℂ and γ([a, b[) ⊆ A ⊆ (~)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 eit of the circle line λ with the length 2πr for t ∈ [0, 2π[ and 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 ∈ (~)ℂ and r ∈ (~)>0, we obtain, provided f: Br(c) → (~)ℂ satisfies the condition of the integral formula,

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 → (~)ℂ with A ⊆ (~)ℂ and

and n ∈ ℕ, ak, c, ajk, cj(~)ℂ as well as pairwisely different cj ≠ c is called the residue resc f.

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

with n ∈ ℕ, ajk, cj(~)ℂ and cj pairwisely different, it applies

for a closed path γ: [a, b[ → (~)ℂ.

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

and

Intermediate value theorem: Let be f: [a, b] → (~)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 ℝ, it maps to any value of cℝ 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 cℝ, 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. A derivative can then be defined there as 0.

Proof: Clear from the definitions.

Remark: The following considerations can be transferred analogously to the conventional complex and omit the specification of set of all neighbourhood relations, since they can be easily added.

Product, quotient and chain rule: Let f and g be right-sided (left-sided) exactly differentiable functions and all quotients well defined. Then it applies:

(fg)′r(x) = f′r(x)g(x) + f(↷x) g′r(x),

and

f(g(x))′r = χr(x) f′r(g(x)) g′r(x)

with

Left-sided applies analogously:

(fg)′l(x) = f′l(x)g(↶x) + f(x)g′l(x),

and

f(g(x))′l = χl(x) f′l(g(x)) g′l(x).

with

It applies χr(x) = χl(x) = 1, iff f(g(x)), f(g(↷x)) and f(↷g(x)) resp. f(g(x)), f(g(↶x)) and f(↶g(x)) are lying on a straight line.

Proof: Product and quotient rule are easy to recalculate. It applies:

Thus

The last sentence is valid, because the differences of the f-values are lying anyway on a straight line and, divided by the differences of the corresponding g-values, build a quotient of slopes of two straight lines that share a point. Iff the slopes are equal, the quotient becomes 1, applies therefore the conventional chain rule exactly and it follows the assertion.

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 x0(~)ℝ (i.e. we can set α small enough). We can say analogous from infinitesimal arguments. In order that the chain rule applies sufficiently precise, we must have

with (α-) continuous f and g, thus f(↷g(x)) ≈ f(g(↷x)) and ↷g(x) ≈ g(↷x). We can transfer this into the complex analogously. We can see from the functions f(y) = y±2 and y = g(x) = x2 mit x0 = d0 and y ∈ (~)ℝ that χ can attain almost any value in (~)ℝ. If f is not affine-linear or g not the identity or a translation, it is quite unlikely that the three f-values are lying on a straight line.

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

f-1r(y) = 1/f′r(x) bzw. f-1l(y) = 1/f′l(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 ∈ (~)ℝ as well as f(↷w) and g(↷w) not both simultaneously 0, and analogously left-sided from

Definition: Let be f: A → (~)ℍ with A ⊆ (~)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.

Exchange theorem: The result of multiple partial derivatives of a function f: A → (~)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 (trans-natural) mathematical induction.

Example: Let be f: ℝ2 → ℝ given by f(0, 0) = 0 and f(x, y) = xy3/(x2 + y2) 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 → (~)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 ⊆ (~)ℂ, the neighbourhood relation B ⊆ A2 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 (trans-)real-valued function with arguments ∈ (~)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). The h-homogeneous set A ⊆ (~)n is called h-convex if A contains also the vertices of each maximum-dimensional h-cube wj of (~)n, with j ∈ ~ℕ*, of the edge length h, whose centre is located on the distance between two arbitrary different points ∈ (~)n \ ∂A from h-cubes wk of (~)n with k ∈ ~ℕ* whose vertices are in A and where the coordinates of all vertices are a integer multiple of h. Here we assume that we can divide h into at least three (ideal) parts.

Definition: A is called star h-domain if each (ideal) centre of the mentioned h-cubes but one can be connected with the (ideal) centre of the latter by a distance so that no (ideal) point of the latter is outside of the mentioned h-cubes. A is called a simply connected h-set if each closed path through the (ideal) centres of some of the mentioned h-cubes, which is in their interior, can be so diminished by successively replacing an (ideal) centre by the (ideal) centre of a adjacent h-cube that it remains closed and in the interior of the remaining h-cubes involved, until it consists only of an (ideal) centre of an h-cube.

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 ⊆ (~)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, with ↷B γ(t) = γ( ↷D t), t ∈ [a, b[, D ⊆ [a, b]2 and 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 ∈ (~)ℝ, f, g : ∂A → (~)ℝ convex or concave with f < g, 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 if we connect each time the minimum or maximum resp. two arbitrary points else of f and g by maximally two vertical parts of a path and a horizontal one connecting them (maybe ideal) in A, which are in each case to be traversed in both directions. 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.

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 = [a1, b1]×…×[an, bn] ⊆ (~)n with functions fk: C → (~)ℝ, where

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

Goursat's integral lemma: If f is holomorphic in a triangle Δ ⊆ (~)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 Δs ⊆ Δ we must have w.l.o.g. either

or

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 (~)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 trans-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 ⊆ ℂ, infinitesimal h as well as a holomorphic function f: A → ℂ and a closed path γ: [a, b[→ ∂A with ↷B γ(t) = γ(↷D t), 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 ∈ (~)ℂ, it exists a z ∈ (~)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 ∈ (~)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 ∈ (~)>0, thus f(0) = O(d0) in contradiction to the presupposition.

Let be all conventionally complex functions f: A → (~)ℂ for A ⊆ (~)ℂ implicitly so defined that we have max |f(z)| ≤ r := max (~)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 (~)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 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 (trans-) 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 cℂ onto the complex unit disk đd, very condensed from the complex unit disk đ, with |đd| = |cℂ| ≫ |đ|. Therewith, the Riemann mapping theorem is also valid for ℂ. The complete cℂ could not be mapped this way, of course.

Definition: A point z0 ∈ M ⊆ (~)n resp. concerning a sequence (ak) with ak(~)n and (trans-) natural k is called actual α-accumulation point of M resp. of the sequence, if there are in the ball Bα(z0) ⊆ (~)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 ∈ ℕ for z ∈ ℂ an infinite product with pairwise distinct zeros ck ∈ B1/|ℕ*|(0) ⊂ ℂ (disks around 0 with radius 1/|ℕ*|), which are chosen so that |f(ck)| < 1/|ℕ*| applies, for a in a domain G ⊆ ℂ holomorphic function f with f(0) = 0. G contains B1/|ℕ*|(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/|ℕ*|(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 trans-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, the theorems of Picard, since they miss at least n - 1 values in ℂ.

If the Riemann zeta function is for s ∈ ℂ \ {1} defined as

so the integral is, with infinitesimal δ ∈ ℝ>0, not equivalent to

since the branch of the logarithm must not be chosen arbitrarily, if a correct result is expected. Since x is to be integrated in the left integral infinitesimally above, and in the right one infinitesimally below the real axis, the sum of both becomes singular in the points x = δ ± iε, with infinitesimal ε ∈ ℝ>0 for Re s ≤ 1 - even if -iπ is corrected to iπ. This is not astonishing, since nothing has been achieved except a wrong transformation. If δ would be concrete, then the Riemann zeta function for Re s ≤ 1 would not be a holomorphic extension of the one for Re s > 1, and the further reasoning would become incorrect, since the paths determined by x = |δ| would have to be replaced by a path via the points iε, iε - δ, -iε - δ and -iε. Therewith, the Riemann hypothesis has the same quality as the Riemann series theorem - also considering other representations of the Riemann zeta function (known to me).

Remark: First the nonstandard analysis made such an easy demystification possible by having used concrete and infinitesimal quantities gainfully. Perhaps it achieves also by it a condign place in mathematics.

Remark: We can approximate ζ(2n+1) for n ∈ 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 ∈ ℂ \ ℕ* by z and the difference to ln |ℕ|. 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 ∈ ℕ*, with complex rational coefficients, can be represented, except a predetermined conventionally real precision, as the sum of ζ(2n) values, with n ∈ ℕ*, and F′(z) values, with z ∈ 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 Bk and Faulhaber's formula as well as k ∈ ℕ, m, n ∈ ℕ* (generalisation to greater numbers possible):

We obtain from the identity

for real or complex x with (-x)k := (-1)k xk 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 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 ∈ ℕ* and subsequent differentiation.