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REVIEW 3 major objections 5 minor 29 references

Euler and the Gammafunction

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Read through modern notation, Euler's 18th-century papers on the factorial already contain the Gamma-function results later credited to Gauß, Weierstraß, and Mellin.

desk verdict A useful but uneven Euler sourcebook: the translations and the correction of E189 are worth having, while the 'Euler anticipated Mellin' claim overreaches the evidence. read the letter →

arxiv 1908.01571 v5 pith:RS5I5QBE submitted 2019-08-05 math.HO

Alexander Aycock This is my paper · ORCID
classification math.HO MSC 33B1501A5044A15
keywords GammafunctionEulerhistoryofmathematicsMellintransformdifferenceequationsfactorialinterpolationEuler-MaclaurinformulaStirling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is a historical reconstruction with a thesis: Euler's scattered 18th-century papers on interpolating the factorial, read through modern notation, already contain the central results of the modern theory of the Gamma function. The author argues that Euler derived several expressions 'usually attributed to others' — the product formula credited to Gauß, the Weierstraß product expansion, and the integral-representation method that amounts to the Mellin transform — and that Euler's results obtained through his 'moment ansatz' can be justified rigorously by Mellin-transform theory. If the reconstruction is right, textbook attributions should shift modestly and Euler's heuristic art of finding becomes a defensible route into the theory rather than a historical curiosity. The paper also corrects a genuine error in Euler's work: his solution of the difference equation $f(x+1)-f(x)=g(x)$ via differential equations of infinite order misses the term $-\tfrac12 g(x)$, and the correction is shown to yield the Stirling formula for the factorial.

What carries the argument

The load-bearing object is the functional equation $\Gamma(x+1)=x\Gamma(x)$ treated as a difference equation, attacked by three heuristic methods: the 'moment ansatz' (writing the solution as an $x$-th moment $\int t^{x-1}P(t)\,dt$ and deriving a first-order differential equation for $P(t)$), the conversion of the difference equation into a differential equation of infinite order via Taylor's theorem, and the Euler-Maclaurin summation formula. The moment ansatz is the paper's named centerpiece: it is a precursor of the Mellin transform exactly because the Mellin transform converts differential equations into difference equations, which is what the ansatz does in reverse, and the paper shows Euler could even recover the limits of integration from the vanishing of the boundary term. The correction of Euler's infinite-order method turns on the partial fraction decomposition $\frac{1}{e^z-1} = -\frac12 + \frac1z + \sum_{k\neq 0}\frac{1}{z-2k\pi i}$; the constant term $-\tfrac12$ is precisely the term Euler missed, and it is what restores the correct solution and the correct Stirling constant $\sqrt{2\pi}$.

What would settle it

Checking Euler's original Latin texts would settle the historical side: in [E189] one can look for whether Euler's claimed solution of $f(x+1)-f(x)=g(x)$ really omits the $-\tfrac12 g(x)$ term, and in [E123] sections 49-53 one can check whether the moment ansatz appears there in recognizable form or only under the paper's modern reading. On the mathematical side, applying the paper's corrected Fourier solution to $g(x)=\log x$ should reproduce the Stirling expansion of $\log\Gamma(x)$ exactly; if the asymptotic series does not match, the correction's application to the factorial fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that Euler, attacking the interpolation of the factorial $n!$ through the functional equation $\Gamma(x+1)=x\Gamma(x)$, already hit on what later became the Gauß product formula, the Weierstraß product expansion, and the Mellin transform. The 'moment ansatz' — assuming a solution of a difference equation has the form $\int t^{x-1}P(t)\,dt$, deriving a differential equation for $P(t)$, and fixing the integration limits by requiring a boundary term to vanish — is argued in section 1.5.4.7 to be 'actually the same idea' as Mellin's transform, discovered from the side of difference equations rather than hypergeometric differential equations. On the historical side, the paper claims priority for Euler over Gauß for the product representation of the factorial, traces the Weierstraß product construction back to Euler's interpolation theory, and shows that several results Euler stated without proof have rigorous proofs within his reach. On the technical side, the paper identifies and corrects an error in Euler's 1753 paper [E189]: solving $f(x+1)-f(x)=g(x)$ through the zeros of $e^z-1$ drops the term $-\tfrac12 g(x)$; the corrected solution, built on the partial fraction decomposition of $1/(e^z-1)$ and Fourier analysis, recovers the Stirling formula for the factorial.

Load-bearing premise

The anticipation thesis rests on the premise that the paper's modern reconstructions faithfully capture how Euler actually reasoned; if the moment ansatz and the other methods are read into Euler's terse Latin arguments rather than found there, the priority claims overcredit him.

Editorial extensions

If this is right

  • Textbook attributions of the Gaußian product formula and the Weierstraß product expansion would shift toward Euler, who stated or proved them decades before their usual namesakes.
  • Euler's heuristics become a legitimate route into the theory: the moment ansatz is justified rigorously by Mellin-transform theory, so the integral representation $\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,dt$ can be derived from the functional equation instead of being introduced by fiat.
  • The corrected solution of $f(x+1)-f(x)=g(x)$ makes Euler's infinite-order method sound and recovers the Stirling formula $x! \sim \sqrt{2\pi x}\,x^x e^{-x}$, with the constant fixed through the Wallis product.
  • The classification theorems (Bohr-Mollerup, Wielandt) retroactively justify Euler's working assumption that his many different expressions for the factorial are the same function, since each expression can be checked against the same three defining properties.
  • Euler's generalized factorials receive complete Stirling-type asymptotic expansions, with all constants evaluated in terms of the Gamma function and Legendre's duplication formula — steps the paper argues Euler himself could have taken.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the reconstruction is faithful, the same method of reading could be tested on Euler's other heuristic achievements — for instance his functional equation for the zeta function — to see whether this anticipation pattern generalizes beyond the Gamma function.
  • The corrected formula $f(x)=\int^x g(t)\,dt-\tfrac12 g(x)+\sum_{k\neq0}e^{2k\pi i x}\int^x e^{-2k\pi i t}g(t)\,dt$ is a close cousin of the Euler-Maclaurin summation formula; a natural extension would be to prove it in larger function classes than the Schwartz space the paper assumes, which would make Euler's formal operator calculus rigorous where he left it heuristic.
  • A didactic consequence the paper leaves implicit: the Gamma function could be introduced from its functional equation through the moment ansatz, turning Euler's path into a systematic way to discover — not just verify — integral representations for special functions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript is a very long historical and mathematical survey of Euler's work on the gamma function. It reconstructs Euler's derivations in modern notation, supplies English translations of three of Euler's papers (E19, E368, E421), and argues that Euler's moment ansatz anticipates the Mellin transform, that Euler's infinite-order differential-equation technique can be corrected by Fourier analysis, and that results usually attributed to Gauß, Weierstraß, Bohr, Mollerup, and Mellin were already present in Euler. It also proves standard classification theorems, the Weierstraß product expansion, the reflection formula, and the multiplication formula, and it discusses the beta function and Stirling's formula.

Significance. If the historical claims were fully supported, this paper would be a valuable corrective to standard textbook attributions. The mathematical reconstructions are mostly sound: the moment-ansatz derivation of the integral representation is coherent and is checked against Wielandt's theorem, the corrected difference-equation formula in section 1.6 is a useful modern formulation, and the Bohr-Mollerup and Wielandt proofs are standard. The translations of the three Euler papers are a concrete scholarly asset, and the paper honestly flags several places where its historical reconstruction is speculative. However, the central historical conclusion is stronger than the source evidence shown, and one of the technical claims needs a function-space correction.

major comments (3)
  1. [§1.5.4.7, §1.5.2] The load-bearing historical claim that Euler's moment ansatz is 'actually the same idea' as the Mellin transform is not established by the evidence presented. The reconstruction in section 1.5.2 consists of the integral ansatz f(x)=∫t^{x-1}P(t)dt, the auxiliary boundary term t^xQ(t), differentiation and coefficient comparison, and the determination of limits from t^xQ(t)=0; these moves are attributed to E123 §§49-53 and E594 §13, but no line-by-line comparison shows which of them are explicit in Euler's Latin text and which are the author's modern elaboration. The paper itself undercuts the stronger attribution in section 1.5.3.7, where it concedes that Euler never wrote the Eulerian integral representation of the hypergeometric function even though he had all ingredients. 'Follows from Euler's work' is not the same as 'Euler's idea'. I recommend revising the priority claims to say that Euler's procedure can be viewed as a precursor of the Mellin transform, and adding a source-critical discussion of which reconstructed steps are original.
  2. [§1.6.3.2, Theorem 1.6.7, §1.6.3.5] The statement of Theorem 1.6.7 is not correct as a solution theorem for general Schwartz data. The theorem says 'with f and g∈S(R)' and then constructs a particular solution by indefinite integrals and an infinite sum over l. But if a Schwartz solution exists, Fourier transformation gives (Σ a_k e^{ipk}) f̂(p)=ĝ(p), so ĝ must vanish at every zero p_l of the symbol; the theorem does not state this compatibility condition. Conversely, for the simple equation f(x+1)-f(x)=g(x), the term ∫^x g(y)dy in the displayed formula is not in S(R) when ĝ(0)≠0, since it tends to a nonzero constant. Thus the formula does not solve arbitrary g∈S(R) within S(R). The authors should either impose the necessary vanishing conditions on ĝ, or state the result in a function space that accommodates the periodic terms, or reformulate the theorem as: if a Schwartz solution exists, it has this form. This is load-bearing because section 1.6 claims to give a rigorous correction of Euler's approach.
  3. [§1.2.2, §1.5.3.7, §1.10] The paper's own internal caveats are not consistently carried into its conclusions. Section 1.2.2 explicitly says 'It is speculation, whether Euler found his result influenced by Bernoulli's formula', and section 1.5.3.7 concedes that the hypergeometric integral representation is not found in any of Euler's works. Yet the abstract and the concluding overview present the anticipation claims as established facts, and section 1.5.4.7 asserts that 'Euler's results are indeed correct' and that the moment method 'is actually the same idea' as the Mellin transform. Since the paper's contribution is primarily historical attribution, these caveats should be reflected in the final claims; otherwise the reader cannot distinguish mathematically valid reconstructions from historically documented discoveries.
minor comments (5)
  1. [§1.4.4] In the proof of the Weierstraß product expansion, the text says 'log Γ(1) = 1'; it should be 'log Γ(1) = 0' before concluding Γ(1)=1.
  2. [§1.5.2.1, Theorem 1.5.1] The boundedness check for Wielandt's theorem writes '|Γ(x)| ≤ Re(Γ(x))' for Re x>0, which is false for non-real x; the intended estimate is |Γ(x+iy)| ≤ Γ(Re(x+iy)) = Γ(x) for x>0.
  3. [§1.6.3.5] There are several typos in the summation ranges, for example 'k∈Z∈{0}' should be 'k∈Z\{0}' and the formula '∑_{l=∞}' should be '∑_{l=-∞}^{∞}'.
  4. [§1.7.3.1] The name 'Carmichel' is misspelled; it should be 'Carmichael' in the text and in the citation [Ca36].
  5. [§1.5.5] The long intersection-theory section is mathematically interesting but is not tied to Euler's own texts and interrupts the historical narrative; it could be substantially shortened or moved to an appendix without affecting the main claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's reconstructions are validated against independent external theorems (Wielandt, Bohr-Mollerup, Wallis product, Legendre duplication), and its priority claims are historical interpretations rather than derived predictions.

full rationale

The paper is a historical-expository review. Its load-bearing mathematical derivations are self-contained in the sense required by the circularity standard: the integral representation of the Gamma function is obtained from the functional equation via the moment ansatz and then checked against Wielandt's theorem (Theorem 1.5.1), an external classification theorem stated and proved in Section 1.3.2.1; the Weierstrass product is verified through the Bohr-Mollerup theorem (Section 1.4.4); the Stirling formula is derived from the corrected difference-equation solution with the periodic function determined by the Wallis product (Section 1.6.4.1); and the generalized-factorial constants are evaluated using the integral representation of the Gamma function and Legendre's duplication formula (Sections 1.6.5.2–1.6.5.4). None of these steps fits a parameter to the very result it claims to predict, and none cites the present author as the source of a load-bearing uniqueness theorem. The assertions that Euler 'anticipated' the Mellin transform or that his moment ansatz is 'actually the same idea' (Section 1.5.4.7) are interpretive claims about intellectual priority and mathematical equivalence; they do not reduce a derivation to its own conclusion. The paper itself flags the relevant historical uncertainty ('It is speculation, whether Euler found his result influenced by Bernoulli's formula') and even concedes that Euler never stated the hypergeometric integral representation (Section 1.5.3.7), which further shows that the attribution claims are not being used to force a mathematical result into existence. The only self-referential element is the note that the author has translated Euler's Latin papers; this is a statement of scholarly provenance, not a load-bearing mathematical premise.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The derivations rest on standard pillars of 19th and 20th century analysis that the paper cites and sometimes proves: the Bohr-Mollerup theorem (proved in 1.3.2.2), Wielandt's theorem (proved in 1.3.2.1), Fourier inversion and convolution on the Schwartz space (1.6.3), the partial fraction expansion of 1/(e^z-1) (used in 1.6.2.3, proved in appendix 1.11.2), the Euler-Maclaurin formula (1.7), and twisted de Rham and intersection-theory results imported from [Ao11] and [Ch95] (1.5.5). No free parameters are fitted to data, and no new entities are postulated. The only non-mathematical premise is a historical-interpretive one about what Euler's texts mean, which the paper repeatedly flags as speculation.

assumptions (8)
  • standard math Bohr-Mollerup theorem: a positive, logarithmically convex function on the positive reals with f(x+1)=x f(x) and f(1)=1 is the Gamma function.
    Used as the classification anchor to identify product representations and the corrected Stirling result with Γ(x). Proved in section 1.3.2.2 and invoked in sections 1.4.4 and 1.6.4.1.
  • standard math Wielandt's theorem: a function holomorphic in a strip 1≤Re z<2, bounded there, and satisfying f(z+1)=z f(z) is a constant multiple of Γ(z).
    Used to validate the moment-ansatz integral representation as the true Γ-function in section 1.5.2.1. Proved in section 1.3.2.1.
  • standard math Fourier inversion and convolution theorems on the Schwartz space S(R).
    Basis of the corrected solution of f(x+1)-f(x)=g(x) in Theorem 1.6.7, section 1.6.3; the solution is derived in this function class.
  • standard math Partial fraction expansion 1/(e^z-1) = -1/2 + 1/z + Σ_{k≠0} 1/(z-2kπi).
    The load-bearing identity in the correction of Euler's E189 approach, section 1.6.2.3, with proof deferred to appendix 1.11.2.
  • standard math Euler-Maclaurin summation formula and the Bernoulli-number generating function z/(e^z-1).
    The framework of section 1.7, used to derive Stirling's formula and the generalized-factorial asymptotics in section 1.6.5.
  • standard math Twisted (co)homology and intersection-theory theorems for hypergeometric integrals as developed in Aomoto, Cho, and Frellesvig.
    Section 1.5.5 states twisted de Rham duality, Stokes' theorem for local systems, and the decomposition formula without proof, citing [Ao11], [Ch95], and [Fr19].
  • standard math Analytic continuation of the Γ-integral to C minus the non-positive integers, with simple poles of residue (-1)^n/n!.
    Used in the Weierstrass product proof and in Wielandt's theorem, section 1.3.1.2.
  • domain assumption Historical inference that Euler knew Bernoulli's formula and that Euler's preserved texts reveal his method.
    Section 1.2.2 infers Euler's knowledge from a footnote in Juskevic's correspondence edition and from a sentence in Euler's letter to Goldbach. The paper itself calls parts of this 'speculation.' This premise is load-bearing for the anticipation claims.

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Cite this review

Pith. "Pith review of Euler and the Gammafunction." pith.science (2026). https://pith.science/paper/RS5I5QBE

@misc{pith2026190801571,
  author       = {Pith},
  title        = {Pith review of: Euler and the Gammafunction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RS5I5QBE}},
  note         = {Machine review of arXiv:1908.01571}
}
read the original abstract

We review Euler's idea on the Gammafunction. We will explain, how Euler obtained them and how Euler's ideas anticipate more modern approaches and theories. Furthermore, some questions asked by Euler are answered.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    one will have the ordinates y··· 1, 1, 2, 6, 24, 120, 720 etc

    While the letter x denotes the abscissa and y the ordinate, this equation immediately indicates the quantity only of those ordinates corresponding to integer numbers; for, if one had the abscissas x··· 0, 1, 2, 3, 4, 5, 6 etc. one will have the ordinates y··· 1, 1, 2, 6, 24, 120, 720 etc. such that, while the abscissas are taken according to the natural n...

    work page

  2. [2]

    hypergeometric

    But except for these ordinates corresponding to abscissas expressed by integer numbers, those are especially noteworthy, which fall into the middle between them from the equation; and they are all determined by the one I once showed to correspond to the abscissa x = 1 2 and to be equal to 1 2 √π. Therefore, since √ π = 1.77245385090548, all these ordinate...

    work page arXiv

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    Questions of this kind first concern the determination of the remaining points of a curve in addition to those which are easily assigned

    The consideration of this curve raises many rather curious questions, providing a reason to examine it more accurately; and their solution seem to be even more interesting, since the equation for our curve cannot be expressed in the usual manner. Questions of this kind first concern the determination of the remaining points of a curve in addition to those ...

    work page

  4. [4]

    the factors of which must be continued to infinity

    Since the propounded equation y = 1· 2· 3··· x can only hold, if x is an integer number, it must be cast into another form which is not restricted by this condition; this can be achieved in multiple ways using expressions running to infinity, among which at first this one occurs: y = 1 1 + x (2 1 )x · 2 2 + x (3 2 )x · 3 3 + x (4 3 )x · 4 4 + x (5 4 )x · et...

    work page

  5. [5]

    This formula can be generalised a bit; for, since the whole task reduces to this that the factor (n + 1)x becomes equal to the last denominator (n + 1)(n + 2)(n + 3)··· (n + x), in the case, in which n is an infinite number, it is evident that this condition is also satisfied, if the factor is in general set to (n + a)x, where a is an arbitrary finite number...

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    are easily derived

    But this expression is the more useful, the faster the factors converge to one, which happens by taking a = 1+x 2 ; indeed, then the calculation will become as much easier as smaller numbers are substituted for x; but it always suffices to have investigated ordinates for abscissas x between one and zero, since from there the ordinates corresponding to x + ...

    work page

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    But the calculation is executed more conveniently, if our expression is terminated at each factor; for, then the following formulas coming continuously closer to the truth will result: y = 1 1 + x (3 + x 2 )x y = 1 1 + x· 2 2 + x (5 + x 2 )x y = 1 1 + x· 2 2 + x· 3 3 + x (7 + x 2 )x y = 1 1 + x· 2 2 + x· 3 3 + x· 4 4 + x (9 + x 2 )x y = 1 1 + x· 2 2 + x· ...

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    − log(1 + x)− log ( 1 + x 2 ) − log ( 1 + x 3 ) − log ( 1 + x 4 ) − etc

    But products of this kind are expanded most conveniently using logarithms; and first from the general formula involving the arbitrary number a we obtain: log y = x log a + x log a + 1 a + x log a + 2 a + 1 + x log a + 3 a + 2 + x log a + 4 a + 3 + etc. − log(1 + x)− log ( 1 + x 2 ) − log ( 1 + x 3 ) − log ( 1 + x 4 ) − etc. having taken a = 1+x 2 this seri...

    work page

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    Let us take a definite number of terms of the first series, which number we want to be = n, and since the upper part is reduced to the single term x log(a + n), it will be log y = x log(a + x)− log(1 + x)− log ( 1 + 1 2 x ) − log ( 1 + 1 3 x ) −···− log ( 1 + 1 n x ) , which exp...

  2. [10]

    But except for those formulas, in which the ordinate y corresponding to a certain abscissa x is assigned, my method to sum progressions indefinitely 4 provides us with an extraordinary expression accommodated to our purposes. For, since log y = log 1 + log 2 + log 3 + log 4 +··...

  3. [11]

    Or, if one prefers the exponential form, it will also be y = ∫ e−vvpdv, extending the integration from v = 0 to v = ∞

    Finally, the ordinate y can even be exhibited by an integral formula; for, having put the abscissa x = p and having introduced the new variable u, independent of the quantity p, the ordinate will be y = ∫ du ( log 1 u )p , if the integration is extended from the value u = 0 to...

  4. [12]

    y = 1 1 + x ( 2 1 )x · 2 2 + x ( 3 2 )x · 3 3 + x ( 4 3 )x · 4 4 + x ( 5 4 )x · etc

    Therefore, lo and behold the many solutions of our first question, in which for an arbitrary abscissa x, even though it is expressed by a non-integer number, the value of the ordinate y was sought after; it will be helpful to have listed up the principal ones, that from here in...

  5. [13]

    To this end, formula V seems especially suitable, from which we conclude: dy ydx =−∆ + 1 + 1 2 + 1 3 + 1 4 + etc

    Therefore, here we assume that for the abscissa x the value of the ordinate y has already been found, and since the direction of the tangent is defined by the ratio of the differentials dy dx , by which fraction the tangent of the angle, in which the tangent at that point is in...

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    Therefore, first let us define the tangents for the abscissas x which are expressed by positive numbers, since the ordinates y are given. I. Therefore, let x = 0 and, because of y = 1, dy dx =−∆ =−0.5772156649 = tan ϕ, whence the angle ϕ =−29◦59′29′′, where the sign − indicates ...

  7. [15]

    Let x = 1 2, it will be y = 1 2 √π and dy ydx =−∆ + 1− 2 3 + 1 2− 2 5 + 1 3− 2 7 + etc

    Hence let us also define the tangents for the intermediate points, and first certainly for those corresponding to positive abscissas: I. Let x = 1 2, it will be y = 1 2 √π and dy ydx =−∆ + 1− 2 3 + 1 2− 2 5 + 1 3− 2 7 + etc. or dy ydx =−∆ + 2 (1 2− 1 3 + 1 4− 1 5 + etc. ) =−∆ + ...

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    Before I proceed, I observe, if for any abscissa it was x = p, y = q, tan ϕ = r, that then for the following abscissa it will be x = p + 1, y = q(p + 1) and tan ϕ = r(p + 1) + q, Translation of E368 189 but for the preceding one x = p− 1, y = q p and tan ϕ = r p− q pp , whence...

  9. [17]

    The same differential equation serves for finding the point µ of the curve, where the ordinate is the smallest or the tangent is parallel to the axis. Therefore, having put dy dx = 0, the corresponding abscissa x must be found from this equation: ∆ = x 1 + x + x 2(2 + x) + x 3(...

  10. [18]

    Therefore, since according to formula V log q =−∆p + p + 1 2 p + 1 3 p + 1 4 p + etc

    Therefore, for the given abscissa x = p let the ordinate y = q have been found; and now one has to find the ordinate, which corresponds to the abscissa p + ω differing from that one by just a small amount. Therefore, since according to formula V log q =−∆p + p + 1 2 p + 1 3 p +...

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    Here the coordinates p and q can be considered as constants, since the letters ω and ψ denote two new coordinates taken from a given point of the curve and parallel to the first set; from their relation defined here the nature of the curve around that point is easily investigate...

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    = P, 1 (1 + p)2 + 1 (2 + p)2 + 1 (3 + p)2 + 1 (4 + p)2 + etc

    But in order to extend the investigation of the direction and the curvature from the principal point defined by the coordinates p and q to the points of the curve, for the sake of brevity, let us set −∆ + p 1 + p + p 2(2 + p) + p 3(3 + p) + p 4(4 + p) + etc. = P, 1 (1 + p)2 + 1...

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    = 2(1− log 2)− ∆ = 0.03648997397857 Q = 4 32 + 4 52 + 4 72 + 4 92 + etc

    Since this point is not far away from the point corresponding to the abscissa = 1 2 and the ordinate = 1 2 √π, let us set p = 1 2 so that q = 1 2 √π, and hence first let us find the values of the letters P, Q, R, S etc., which will result as: P = − ∆ + 1 3 + 1 2· 5 + 1 3· 7 + et...

  14. [22]

    Hence let us especially define the point µ, where the ordinate is the smallest simple approxi- mations shows it to correspond to the abscissa x = 0.4616, having set Translation of E368 195 p + ω = 1 2 + ω = 0.4616, one finds approximately ω =−0.0383, which value must be investig...

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    Now let us in general differentiate to define the value of ψ from the logarithmic equation, and, having done the calculation, we will obtain: ψ q = + 0.0364899740ω + 0.468066860ω2 − 0.121069221ω3 + 0.16321479ω4 − 0.09360753ω5 + etc., which terms suffice, if the value of ω is ver...

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    I investigated these determinations of the lowest point µ of the curve with all eagerness such that it cannot without any reason be conjectured, as this point has an extraordinary property, that the numbers exhibiting its nature contain in this way a certain elegance, and if t...

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    Before I end this speculation, it will helpful to have observed that the formula 1· 2· 3··· x can also be expressed indefinitely in terms of the following series xx− x(x− 1)x + x(x− 1) 1· 2 (x− 2)x− x(x− 1)(x− 2) 1· 2· 3 (x− 3)x + etc., which, as often as x is a positive intege...

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    Since for smaller exponents x the matter is obvious, I reason as follows, i.e

    These are certainly obvious from the results demonstrated about the difference of each order of algebraic progressions, but nevertheless from the nature of these series the truth is not easily uncovered; thus, the following proof seems to be in order. Since for smaller exponen...

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    ’De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt

    But although this expression is sufficiently elegant and worth one’s complete attention, it is nevertheless less useful for our task, in which the hypergeometric curve it propounded, since for the cases, in which x is a fractional number, this series not only runs to infinity bu...

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    Jacob Bernoulli on the other hand also was a teacher of L’Hospital

    He most likely knew the rule from Jacob Bernoulli, his mentor in his teenage years. Jacob Bernoulli on the other hand also was a teacher of L’Hospital. Translation of E19 165 §14 Therefore, because 1− x0 0 =− log(x), it will be (1− x0)n 0n = (− log(x))n and therefore the gener...

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    sinus",

    Vieweg, Braunschweig Wiesbaden 1988 [Wi96] H. Wielandt Mathematische Werke Vol. 2, de Gruyter, Berlin New York 1996 Euler and the Γ-Function 153 [Yo97] M. Yoshida Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces, Vieweg, edition from 1997 Appen...

    1988

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