DOCSLIB.ORG

The Gamma Function

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Gamma Function The Gamma function Marco Bonvini October 9, 2010 1 Gamma function The Euler Gamma function is defined as Z ∞ t z 1 Γ(z) dt e− t − . (1) ≡ 0 It is easy to show that Γ(z) satisfy the recursion relation Γ(z + 1) = z Γ(z) : (2) indeed, integrating by parts, Z ∞ t z Γ(z + 1) = dt e− t 0 Z t z ∞ ∞ t z 1 = e− t + z dt e− t − − 0 0 = z Γ(z) . (3) For z = n integer eq. (2) is the recursion relation of the factorial, and thus we have Γ(n + 1) n! ; (4) ∝ because in addition Γ(1) = 1 (easly derived from the definition), we have the identification Γ(n + 1) = n! , (5) and in this sense the Gamma function is a complex extension of the factorial. The sequence of Gamma function computed in all half-integers can be obtained using subsequently the recursion relation (2) and knowing that 1 Γ = √π (6) 2 that is easy to compute: Z 1 ∞ t 1 Γ = dt e− t− 2 2 0 Z ∞ x2 = 2 dx e− 0 √π = 2 = √π . (7) 2 1 15 10 5 !2 2 4 !5 !10 Figure 1: Re Γ(x) for real x. 1.1 Analytical structure First, from the definition (1), we see that Γ(z¯) = Γ(z) , (8) that is to say that Gamma is a real function, and in particular Im Γ(x) = 0 for x R. ∈ Inverting eq. (2) we have 1 Γ(z) = Γ(z + 1) , (9) z and when z = 0 it diverges (because Γ(1) = 1 is finite). It is evident that z = 0 corresponds to a simple pole of order 1 and residue 1: indeed Res Γ(z) = lim z Γ(z) = lim Γ(z + 1) = 1 . (10) z=0 z 0 z 0 → → Formally, we can discover that the Gamma function has simple poles in all negative integers simply iterating the recursion (2): 1 Γ( n) = Γ( n + 1) − n − − = ... ( 1)n = − Γ(0) (n N) . (11) n! ∈ ( 1)n This brings to the conclusion that Γ( n) has a simple pole of order 1 and residue − . More − n! rigorously, this can be shown in the usual way: Res Γ(z n) = lim z Γ(z n) z=0 − z 0 − → z = lim Γ(z n + 1) z 0 z n − → − = ... 2 z = lim Γ(z) z 0 (z n)(z n + 1) (z 1) → − − ··· − ( 1)n = − (n N) . (12) n! ∈ In all the complex plane (except the negative real axis) the Gamma function is well defined. Fon non-integer negative real values the Gamma function can be analytically continued (as we have seen for example for half-integers, positives and negatives). We can prove these results in a simpler way. Starting from the definition (1) we rewrite Z Z 1 ∞ t z 1 t z 1 Γ(z) = dt e− t − + dt e− t − 1 0 Z Z 1 X n ∞ t z 1 z 1 ∞ ( 1) n = dt e− t − + dt t − − t n! 1 0 n=0 Z X n ∞ t z 1 ∞ ( 1) 1 = dt e− t − + − (13) n! z + n 1 n=0 where the first integral is an analytic function of z and the second term gives the position of the poles and their residues. Moreover, the Gamma function has an essential singularity to complex infinity, because Γ 1 has a non-defined limit for z 0. This is to say that the Gamma function is not well z → defined in the compactified complex plane. Now we want to show that near a pole in n one has the expansion − ( 1)n 1 Γ(z n) = − + ψ(n + 1) + (z) (14) − n! z O where ψ(z) is the logarithmic derivative of the Gamma function, defined in (41); note that ψ(n + 1) has an explicit expression given in eq. (44). First, note that 1 Γ(z n) = Γ(z n + 1) − z n − − = ... 1 = Γ(z + n + 1) (z n)(z n + 1) (z 1) z (z + 1) (z + n) − − ··· − ··· 1 = G(z, n) (15) z where we have defined Γ(z + n + 1) ( 1)n h i G(z, n) = = Γ(z + n + 1) − 1 + (z2) . (16) (z2 n2) (z2 1) (n!)2 O − ··· − By Taylor expanding G(z, n) around z = 0 in (15) we get 1 h 2 i Γ(z n) = G(0, n) + z G0(0, n) + (z ) − z O ( 1)n 1 1 = − + Γ0(n + 1) + (z) , (17) n! z n! O that is the claimed result. 3 1.2 Alternative definitions There are some alternative definitions of the Gamma function. One, due to Euler, is 1 z 1 Y∞ (1 + ) n!(n + 1)z Γ(z) = n = lim . (18) z 1 + z n z(z + 1) (z + n) n=1 n →∞ ··· It is easy to prove (using the expression with the limit) that with this definition the recursion relation (2) is satisfied, and also that for z = 1 gives Γ(1) = 1. Another one, due to Weierstrass, is h i γz Y z/n n! exp z 1 + 1 + ... + 1 e− ∞ e γz 2 n Γ(z) = = e− lim , (19) z 1 + z n z(z + 1) (z + n) n=1 n →∞ ··· where γ is the Euler-Mascheroni number defined in Section 2.2 The equivalence to the Euler definition can be seen by taking the logarithm of this two forms: the first gives X∞ 1 z log Γ(z) = log z + z log 1 + log 1 + (20) − n − n n=0 and the second gives X∞ z z log Γ(z) = log z γz + log 1 + . (21) − − n − n n=0 Comparing this two expressions we note that they are equal because of eq. (48). Note that this two expression are useful in order to numerically approximate log Γ(z), by truncating the series to a certain value of n. 1.3 Euler reflection formula A useful formula is π Γ(z) Γ(1 z) = (22) − sin(πz) called the Euler reflection formula. To demonstrate it, consider the alternative definition (18) and note that 1 1 = Γ(z) Γ(1 z) z Γ(z) Γ( z) − − − z z z( z) Y∞ (1 + n ) (1 n ) = − z − z z 1 1 − − n=1 (1 + n ) (1 + n ) ! Y∞ z2 = z 1 − n2 n=1 sin(πz) = (23) π where we have used ! Y∞ z2 sin(πz) = πz 1 . (24) − n2 n=1 4 1.4 Stirling approximation The Stirling approximation for the factorial is n n n! = √2πn e− n (25) which is valid for large n. To demonstrate it, we perform a saddle point approximation to the integral (1): Z ∞ t z Γ(z + 1) = dt e− t 0 Z ∞ t+z log t = dt e− 0 now we expand the exponent near its maximum and get Z ∞ 1 dt exp z + z log z t2 ' 0 − − 2z Z z z ∞ 1 t2 e− z dt e− 2z ' −∞ z z = √2πz e− z (26) that is the claimed result. 1.5 Derivative recursion relation Consider the Taylor expansion near z = 0 of Γ(z + z0 + 1): X∞ Γ(k)(z + 1) Γ(z + z + 1) = 0 zk ; (27) 0 k! k=0 using eq. (2) this is equal to (z + z0) Γ(z + z0), that expanded is X∞ Γ(k)(z ) (z + z ) Γ(z + z ) = (z + z ) 0 zk 0 0 0 k! k=0 X k ∞ z h (k) (k 1) i = z Γ(z ) + z Γ (z ) + k Γ − (z ) . (28) 0 0 k! 0 0 0 k=1 Comparing the two expansions, we get the equality (k > 0) (k) (k) (k 1) Γ (z0 + 1) = z0 Γ (z0) + k Γ − (z0) . (29) 1.6 Inverse Gamma function We can define an inverse Gamma function 1 ∆(z) (30) ≡ Γ(z) 5 2.5 2.0 1.5 1.0 0.5 !2 2 4 6 !0.5 !1.0 Figure 2: Re ∆(x) for real x. which is an entire function (analytical in all the complex plane). The recursion relation (2) becomes ∆(z) = z ∆(z + 1) . (31) We are able to produce an integral representation of the ∆ function: consider the Laplace z 1 transform of the function t − with respect to t: Z ∞ st z 1 z dt e− t − = s− Γ(z) (32) 0 z and then we have that the inverse Laplace tranform of s− is Z c+i 1 ∞ ts z z 1 ds e s− = ∆(z) t − (33) 2πi c i − ∞ z where c has to be choosen to the right of all the singularities of s− , that is c > 0. Now, by choosing t = 1, we get Z c+i 1 ∞ s z ∆(z) = ds e s− , (34) 2πi c i − ∞ where the integration path can be deformed to be a contour of the negative real axis. Note that for positive integers z = n + 1 the branch cut vanishes and only a pole in s = 0 remains, and then the path can be closed around s = 0 to obtain I 1 1 s n 1 = ds e s− − , (35) n! 2πi that can be also easly demonstrated using the residue theorem. Now consider the Taylor expansion near z = 0 of ∆(z + z0): X∞ ∆(k)(z ) ∆(z + z ) = 0 zk ; (36) 0 k! k=0 using eq. (31) this is equal to (z + z0) ∆(z + z0 + 1), that expanded is X∞ ∆(k)(z + 1) (z + z ) ∆(z + z + 1) = (z + z ) 0 zk 0 0 0 k! k=0 6 X k ∞ z h (k) (k 1) i = z ∆(z + 1) + z ∆ (z + 1) + k ∆ − (z + 1) .
Load more

Recommended publications
  • Introduction to Analytic Number Theory the Riemann Zeta Function and Its Functional Equation (And a Review of the Gamma Function and Poisson Summation)
    Math 229: Introduction to Analytic Number Theory The Riemann zeta function and its functional equation (and a review of the Gamma function and Poisson summation) Recall Euler’s identity: ∞ ∞ X Y X Y 1 [ζ(s) :=] n−s = p−cps = . (1) 1 − p−s n=1 p prime cp=0 p prime We showed that this holds as an identity between absolutely convergent sums and products for real s > 1. Riemann’s insight was to consider (1) as an identity between functions of a complex variable s. We follow the curious but nearly universal convention of writing the real and imaginary parts of s as σ and t, so s = σ + it. We already observed that for all real n > 0 we have |n−s| = n−σ, because n−s = exp(−s log n) = n−σe−it log n and e−it log n has absolute value 1; and that both sides of (1) converge absolutely in the half-plane σ > 1, and are equal there either by analytic continuation from the real ray t = 0 or by the same proof we used for the real case. Riemann showed that the function ζ(s) extends from that half-plane to a meromorphic function on all of C (the “Riemann zeta function”), analytic except for a simple pole at s = 1. The continuation to σ > 0 is readily obtained from our formula ∞ ∞ 1 X Z n+1 X Z n+1 ζ(s) − = n−s − x−s dx = (n−s − x−s) dx, s − 1 n=1 n n=1 n since for x ∈ [n, n + 1] (n ≥ 1) and σ > 0 we have Z x −s −s −1−s −1−σ |n − x | = s y dy ≤ |s|n n so the formula for ζ(s) − (1/(s − 1)) is a sum of analytic functions converging absolutely in compact subsets of {σ + it : σ > 0} and thus gives an analytic function there.
    [Show full text]
  • INTEGRALS of POWERS of LOGGAMMA 1. Introduction The
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 139, Number 2, February 2011, Pages 535–545 S 0002-9939(2010)10589-0 Article electronically published on August 18, 2010 INTEGRALS OF POWERS OF LOGGAMMA TEWODROS AMDEBERHAN, MARK W. COFFEY, OLIVIER ESPINOSA, CHRISTOPH KOUTSCHAN, DANTE V. MANNA, AND VICTOR H. MOLL (Communicated by Ken Ono) Abstract. Properties of the integral of powers of log Γ(x) from 0 to 1 are con- sidered. Analytic evaluations for the first two powers are presented. Empirical evidence for the cubic case is discussed. 1. Introduction The evaluation of definite integrals is a subject full of interconnections of many parts of mathematics. Since the beginning of Integral Calculus, scientists have developed a large variety of techniques to produce magnificent formulae. A partic- ularly beautiful formula due to J. L. Raabe [12] is 1 Γ(x + t) (1.1) log √ dx = t log t − t, for t ≥ 0, 0 2π which includes the special case 1 √ (1.2) L1 := log Γ(x) dx =log 2π. 0 Here Γ(x)isthegamma function defined by the integral representation ∞ (1.3) Γ(x)= ux−1e−udu, 0 for Re x>0. Raabe’s formula can be obtained from the Hurwitz zeta function ∞ 1 (1.4) ζ(s, q)= (n + q)s n=0 via the integral formula 1 t1−s (1.5) ζ(s, q + t) dq = − 0 s 1 coupled with Lerch’s formula ∂ Γ(q) (1.6) ζ(s, q) =log √ . ∂s s=0 2π An interesting extension of these formulas to the p-adic gamma function has recently appeared in [3].
    [Show full text]
  • Sums of Powers and the Bernoulli Numbers Laura Elizabeth S
    Eastern Illinois University The Keep Masters Theses Student Theses & Publications 1996 Sums of Powers and the Bernoulli Numbers Laura Elizabeth S. Coen Eastern Illinois University This research is a product of the graduate program in Mathematics and Computer Science at Eastern Illinois University. Find out more about the program. Recommended Citation Coen, Laura Elizabeth S., "Sums of Powers and the Bernoulli Numbers" (1996). Masters Theses. 1896. https://thekeep.eiu.edu/theses/1896 This is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Masters Theses by an authorized administrator of The Keep. For more information, please contact [email protected]. THESIS REPRODUCTION CERTIFICATE TO: Graduate Degree Candidates (who have written formal theses) SUBJECT: Permission to Reproduce Theses The University Library is rece1v1ng a number of requests from other institutions asking permission to reproduce dissertations for inclusion in their library holdings. Although no copyright laws are involved, we feel that professional courtesy demands that permission be obtained from the author before we allow theses to be copied. PLEASE SIGN ONE OF THE FOLLOWING STATEMENTS: Booth Library of Eastern Illinois University has my permission to lend my thesis to a reputable college or university for the purpose of copying it for inclusion in that institution's library or research holdings. u Author uate I respectfully request Booth Library of Eastern Illinois University not allow my thesis
    [Show full text]
  • Notes on Riemann's Zeta Function
    NOTES ON RIEMANN’S ZETA FUNCTION DRAGAN MILICIˇ C´ 1. Gamma function 1.1. Definition of the Gamma function. The integral ∞ Γ(z)= tz−1e−tdt Z0 is well-defined and defines a holomorphic function in the right half-plane {z ∈ C | Re z > 0}. This function is Euler’s Gamma function. First, by integration by parts ∞ ∞ ∞ Γ(z +1)= tze−tdt = −tze−t + z tz−1e−t dt = zΓ(z) Z0 0 Z0 for any z in the right half-plane. In particular, for any positive integer n, we have Γ(n) = (n − 1)Γ(n − 1)=(n − 1)!Γ(1). On the other hand, ∞ ∞ Γ(1) = e−tdt = −e−t = 1; Z0 0 and we have the following result. 1.1.1. Lemma. Γ(n) = (n − 1)! for any n ∈ Z. Therefore, we can view the Gamma function as a extension of the factorial. 1.2. Meromorphic continuation. Now we want to show that Γ extends to a meromorphic function in C. We start with a technical lemma. Z ∞ 1.2.1. Lemma. Let cn, n ∈ +, be complex numbers such such that n=0 |cn| converges. Let P S = {−n | n ∈ Z+ and cn 6=0}. Then ∞ c f(z)= n z + n n=0 X converges absolutely for z ∈ C − S and uniformly on bounded subsets of C − S. The function f is a meromorphic function on C with simple poles at the points in S and Res(f, −n)= cn for any −n ∈ S. 1 2 D. MILICIˇ C´ Proof. Clearly, if |z| < R, we have |z + n| ≥ |n − R| for all n ≥ R.
    [Show full text]
  • Special Values of the Gamma Function at CM Points
    Special values of the Gamma function at CM points M. Ram Murty & Chester Weatherby The Ramanujan Journal An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan ISSN 1382-4090 Ramanujan J DOI 10.1007/s11139-013-9531-x 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media New York. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Ramanujan J DOI 10.1007/s11139-013-9531-x Special values of the Gamma function at CM points M. Ram Murty · Chester Weatherby Received: 10 November 2011 / Accepted: 15 October 2013 © Springer Science+Business Media New York 2014 Abstract Little is known about the transcendence of certain values of the Gamma function, Γ(z). In this article, we study values of Γ(z)when Q(z) is an imaginary quadratic field. We also study special values of the digamma function, ψ(z), and the polygamma functions, ψt (z).
    [Show full text]
  • The Riemann Zeta Function and Its Functional Equation (And a Review of the Gamma Function and Poisson Summation)
    Math 259: Introduction to Analytic Number Theory The Riemann zeta function and its functional equation (and a review of the Gamma function and Poisson summation) Recall Euler's identity: 1 1 1 s 0 cps1 [ζ(s) :=] n− = p− = s : (1) X Y X Y 1 p− n=1 p prime @cp=1 A p prime − We showed that this holds as an identity between absolutely convergent sums and products for real s > 1. Riemann's insight was to consider (1) as an identity between functions of a complex variable s. We follow the curious but nearly universal convention of writing the real and imaginary parts of s as σ and t, so s = σ + it: s σ We already observed that for all real n > 0 we have n− = n− , because j j s σ it log n n− = exp( s log n) = n− e − and eit log n has absolute value 1; and that both sides of (1) converge absolutely in the half-plane σ > 1, and are equal there either by analytic continuation from the real ray t = 0 or by the same proof we used for the real case. Riemann showed that the function ζ(s) extends from that half-plane to a meromorphic function on all of C (the \Riemann zeta function"), analytic except for a simple pole at s = 1. The continuation to σ > 0 is readily obtained from our formula n+1 n+1 1 1 s s 1 s s ζ(s) = n− Z x− dx = Z (n− x− ) dx; − s 1 X − X − − n=1 n n=1 n since for x [n; n + 1] (n 1) and σ > 0 we have 2 ≥ x s s 1 s 1 σ n− x− = s Z y− − dy s n− − j − j ≤ j j n so the formula for ζ(s) (1=(s 1)) is a sum of analytic functions converging absolutely in compact subsets− of− σ + it : σ > 0 and thus gives an analytic function there.
    [Show full text]
  • The Gamma Function the Interpolation Problem
    The Interpolation Problem; the Gamma Function The interpolation problem: given a function with values on some discrete set, like the positive integer, then what would the value of the function be defined for all the reals? Euler wanted to do this for the factorial function. He concluded that Z 1 n! = (− ln x)ndx: 0 Observe that (in modern calculation) Z 1 1 − ln x dx = −x ln x + x 0 0 = 1 + lim x ln x x!0+ ln x = 1 + lim 1 x!0+ x 1 x = 1 + lim 1 x!0+ − x2 −x = 1 + lim = 1 − 0 = 1: x!0+ 1 Which is consistent with 1! = 1. Now consider the following integration, Z 1 (− ln x)ndx: 0 We apply the (Leibnitz version) of the integration by parts formula R u dv = uv − R v du. dv = 1 v = x 1 u = (− ln x)n du = n(− ln x)n−1 − dx x Z 1 1 Z 1 n n n−1 (− ln x) dx = x(− ln x) + n (− ln x) dx 0 0 0 Since (in modern notation and using the same limit techniques as above) lim x(ln x)n = 0 x!0+ 1 (Euler would have probably said that 0(ln(0))n = 0.) We have: Z 1 Z 1 (− ln x)ndx = n (− ln x)n−1dx 0 0 which is consistent with our inductive definition of the factorial function: n! = n · (n − 1)!; so for integers the formula clearly works. Then he used the transformation t = − ln x so that x = e−t to obtain the following.
    [Show full text]
  • 1. the Gamma Function 1 1.1. Existence of Γ(S) 1 1.2
    CONTENTS 1. The Gamma Function 1 1.1. Existence of Γ(s) 1 1.2. The Functional Equation of Γ(s) 3 1.3. The Factorial Function and Γ(s) 5 1.4. Special Values of Γ(s) 6 1.5. The Beta Function and the Gamma Function 14 2. Stirling’s Formula 17 2.1. Stirling’s Formula and Probabilities 18 2.2. Stirling’s Formula and Convergence of Series 20 2.3. From Stirling to the Central Limit Theorem 21 2.4. Integral Test and the Poor Man’s Stirling 24 2.5. Elementary Approaches towards Stirling’s Formula 25 2.6. Stationary Phase and Stirling 29 2.7. The Central Limit Theorem and Stirling 30 1. THE GAMMA FUNCTION In this chapter we’ll explore some of the strange and wonderful properties of the Gamma function Γ(s), defined by For s> 0 (or actually (s) > 0), the Gamma function Γ(s) is ℜ ∞ x s 1 ∞ x dx Γ(s) = e− x − dx = e− . x Z0 Z0 There are countless integrals or functions we can define. Just looking at it, there is nothing to make you think it will appear throughout probability and statistics, but it does. We’ll see where it occurs and why, and discuss many of its most important properties. 1.1. Existence of Γ(s). Looking at the definition of Γ(s), it’s natural to ask: Why do we have re- strictions on s? Whenever you are given an integrand, you must make sure it is well-behaved before you can conclude the integral exists.
    [Show full text]
  • Crandall's Computation of the Incomplete Gamma Function And
    Crandall's computation of the incomplete Gamma function and the Hurwitz Zeta function, with applications to Dirichlet L-series D. H. Bailey∗ J. M. Borweiny January 15, 2015 Abstract This paper extends tools developed by Richard Crandall in [16] to provide robust, high-precision methods for computation of the incomplete Gamma function and the Lerch transcendent. We then apply these to the corresponding computation of the Hurwitz zeta function and so of Dirichlet L-series and character polylogarithms. 1 Introduction In this article we continue the research in [5,6,7] and [16] by providing robust, high- precision methods for computation of the incomplete Gamma function and the Lerch tran- scendent. We subsequently apply these to the corresponding computation of the Hurwitz zeta function and of Dirichlet L-series. 1.1 Organization The organization of the paper is as follows. In Section2 we examine derivatives of the classical polylogarithm and zeta function. In Section3 we reintroduce character polyloga- rithms (based on classical Dirichlet characters). In Section4 we reprise work on the Lerch transcendent by Richard Crandall. In Section5 we consider methods for evaluating the incomplete Gamma function, and in Section6 we consider methods for evaluating the Hur- witz zeta function, relying on the methods of the two previous sections. Finally, in Section 7 we make some concluding remarks. Our dear colleague Richard Crandall, with whom we have collaborated in previous papers in this line of research, passed away in December 2012. In this paper, we not only ∗Lawrence Berkeley National Laboratory (retired), Berkeley, CA 94720, and University of California, Davis, Davis, CA 95616, USA.
    [Show full text]
  • Appendix a Properties of the Gamma Functions
    Appendix A Properties of the Gamma Functions We list here some basic properties of the Gamma function (see, e.g., Abramowitz and Stegun (1964)), defined by ∞ Γ (z)= tz−1e−tdt, ∀z ∈ C with Re(z) > 0. (A.1) 0 √ In particular, we have Γ (1)=1andΓ (1/2)= π. • Recursion formula: Γ (z + 1)=zΓ (z), Γ (n + 1)=n!, (A.2) and − / − / Γ (2z)=(2π) 1 222z 1 2Γ (z)Γ (z + 1/2). (A.3) • Connection with binomial coefficient: , - z Γ (z + 1) = . (A.4) w Γ (w + 1)Γ (z − w + 1) • Relation with Beta function: 1 Γ (x)Γ (y) B(x,y)= tx−1(1 −t)y−1dt = , x,y > 0. (A.5) 0 Γ (x + y) In particular, for α,β > −1, 1 1 (1 − x)α(1 + x)β dx = 2α+β +1 tβ (1 −t)αdt −1 0 (A.6) Γ (α + 1)Γ (β + 1) = 2α+β +1 . Γ (α + β + 2) J. Shen et al., Spectral Methods: Algorithms, Analysis and Applications, 415 Springer Series in Computational Mathematics 41, DOI 10.1007/978-3-540-71041-7, c Springer-Verlag Berlin Heidelberg 2011 416 A Properties of the Gamma Functions • Stirling’s formula: √ ! − / − 1 1 − Γ (x)= 2πxx 1 2e x 1 + + + O(x 3) , x 1. (A.7) 12x 288x2 Moreover, we have √ √ + / + / 1 2πnn 1 2 < n!en < 2πnn 1 2 1 + , n ≥ 1. (A.8) 4n Appendix B Essential Mathematical Concepts We provide here some essential mathematical concepts which have been used in the mathematical analysis throughout the book.
    [Show full text]
  • A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions
    axioms Article A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions Maxie D. Schmidt School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30318, USA; [email protected] or [email protected] Received: 23 April 2019; Accepted: 17 May 2019; Published: 19 May 2019 Abstract: The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence’s ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way. Keywords: generating function; series transformation; gamma function; Hankel contour MSC: 05A15; 30E20; 31B10; 11B73 1. Introduction 1.1. Definitions Given a sequence f fngn≥0, we adopt the notation for the respective ordinary generating function (OGF), F(z), and exponential generating function (EGF), Fb(z), of the sequence in some formal indeterminate parameter z 2 C: n F(z) = ∑ fnz (1) n≥0 fn Fb(z) = ∑ zn. n≥0 n! Notice that we can always construct these functions over any sequence f fngn2N and formally perform operations on these functions within the ring of formal power series in z without any considerations on the constraints imposed by the convergence of the underlying series as a complex function of z.
    [Show full text]
  • GAMMA FUNCTION 1. 1.1. History and Motivation. in the Early 16Th
    GAMMA FUNCTION Abstract. In this paper we explore the history and properties of the Gamma function in an analytic number theoretical context. We analyze the behavior of the Gamma function at its critical points and points of discontinuity, and discuss the convergence of the integral. 1. Introduction 1.1. History and Motivation. In the early 16th century, Leonhard Euler and others attempted to expand the domain of the factorial to the real numbers. This cannot be done with elementary functions, however, with the notions of limits and integrals from the calculus, there were a few expressions developed. In 1738, Euler published the following forms of the generalized factorial: Q1 (1+1=k)n ∙ n! = k=1 1+n=k R 1 n ∙ n! = 0 (− log s) ds for n > 0 For the time being, we will refer to these as the product and integral definitions. In the 19th century, Carl Friedrich Gauss rewrote Euler's product definition to extend the domain to the complex plane, rather than simply the real numbers. This expression uses a limiting process on a series of intermediate functions to represent the factorial. r!rx Γ (x) = r x(1 + x)(2 + x) ::: (r + x) Γ(x) = lim Γr(x) r!1 This was also the time period that the notation Γ(x) was applied to the concept. It was named in 1811 by Adrien-Marie Legendre, who also transformed the integral definition in a very simple way to extend its domain: Z 1 n! = (− log s)nds 0 Let t = − log s: s = −et 1 1 dt = − ds = etds s Z 1 Γ(n) = tn−1e−tdt; 0 which is the more common integral definition that we see today.
    [Show full text]