The gamma function is often referred to as the generalized factorial since \\ gamma. This implies that z has a single pole at z 0 with residue 1. The theorem for complex s can then be obtained by analytic continuation. I have tried proving the analytic continuation of the gamma function. Suppose we are given a function f which is analytic in a region d. The functional relation 2 can be used to nd an analytic continuation of the gamma function for rez 0. Lanczos approximation of the classical gamma function. Analytic continuation and the gamma function pdf need help getting started. The functional equation also gives us the analytic continuation of. Pdf a generalization of hypergeometric and gamma functions. This is an important and fascinating function that generalizes factorials from integers to all complex numbers.
Before this, we will state without proof, an essential theorem about the gamma function. Mod01 lec01 analytic functions of a complex variable. Schwarz, felix klein, and others, and hypergeometric functions made their appearance in the early days of algebraic geometry. Analytic continuation of operators operators acting.
Introduction to analytic number theory more about the gamma. Also, we define a jacobi theta function and xi function with the proof of its functional equation and then it will be used to prove the analytic continuation of zeta function as well as its functional equation. We know that in complex analysis 1, functions can be analytic continued from integer points non the real line to complex plane s, eg. Notes on the riemann zeta function january 25, 2007. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. That recursion then extends to a meromorphic function on c, analytic except for simple poles at 0. A generalization of hypergeometric and gamma functions article pdf available in pakistan journal of statistics 292. Gamma function in this section we consider continuing the factorial. If fz is analytic in a domain dand fz is analytic in a domain d0. The growth properties depend on stirlinglaplace asymptotics of the gamma function s, and the phragm enlindel of theorem. The usual differentiation rules apply for analytic functions. An entire chapter is devoted to analytic continuation of the factorials, as well as why the gamma function is defined as it is holders theorem and the bohrmullerup theorem are discussed. We derive this formula in the lemma and theorem below. How does one motivate the analytic continuation of the.
Most of those we use are very well known, but we will provide all the proofs. The gamma function has an analytic continuation to a meromorphic function on c whose only singularities are simple poles at the negative integers. On analytic continuation of various multiple zetafunctions. In the process, we come across functions for which no such extension exists. The maximal analytic continuation of in is unique, but does not always exist. Some results on the derivatives of the gamma and incomplete gamma function for nonpositive integers zhongfeng sun and huizeng qin abstractthis paper is concerned with some recursive re lations of the derivatives of the gamma function and. Then the gamma function z is analytic for rez 1 except z 0. Riemanns second proof of the analytic continuation of the. Except where indicated otherwise principal branches of.
An excellent introduction to this topic can be found in the book the gamma function by james bonnar. This function is a cornerstone of the theory of special functions. D with fz fz in d, then we say that fis an analytic continuation of f. Mathematics university of british columbia, 1999 a thesis submitted in partial fulfillment of the requirements for the degree of doctor of philosophy in the faculty of graduate studies department of mathematics. Introduction to analytic number theory fall 2015 if you find a mistake, omission, etc. Topic notes jeremy orlo analytic continuation and the gamma function. I saw the functional equation and its proof for the riemann zeta function many times, but ususally the books start with, e. The gamma function is a meromorphic function which extends the factorial function to all. The set class of functions holomorphic in g is denoted by hg. The fact that riemann zeta function doesnt have a zero on res 1 is the most crucial step in the proof of the prime number theorem.
Furthermore, the hurwitz zeta function has been used to theoretically. Since it agrees with on b 0 it is an analytic continuation to b 2. In mathematics, the gamma function is one commonly used extension of the factorial function to. The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.
We will also see that an similar property of ls for. Jun 03, 2014 mod04 lec09 analytic continuation and the gamma function part i. Chapter 9 analytic continuation for every complex problem, there is a solution that is simple, neat, and wrong. If f n converges uniformly on every compact closed and bounded subset of dto the limit function f then f is analytic on d. Mod06 lec19 the idea of a direct analytic continuation or. It is one of the most important and ubiquitous special functions in mathematics, with applications in combinatorics, probability, number theory, di erential equations, etc. An analysis of the lanczos gamma approximation by glendon ralph pugh b. If f n converges uniformly on every compact closed and bounded subset of dto the limit function f then f is analytic. The analytic continuation of s is discussed in the next section, and growth properties later. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. The gamma and the beta function as mentioned in the book 1, see page 6, the integral representation 1.
The functional relation 2 can be used to find an analytic continuation of the gamma function for rez. The gamma function has a large variety of properties. The problem of missing terms in term by term integration involving. In addition, barone adesi et al 2 present an analytic continuation of the hurwitz zeta function and provide an application to calculating the pair production rate of dirac particles under certain conditions. A point where the function fails to be analytic, is called a singular point or singularity of the function. This process can be repeated for 2 riemann zeta function. Analytic continuation means extending an analytic function defined in a domain to one defined in a larger domain. The gamma function then is defined as the analytic continuation of this integral. Understanding the analytic continuation of the gamma function. This proof of the analytic continuation is known as the second riemannian proof.
The aim of these lectures is to provide an intorduction to the theory of the riemann zeta function for students who might later want to do research on the subject. Mellin, and then discuss the euler sum and its multivariable generalization. The gamma and zeta functions suppose we are given a function f which is analytic in a region d. A course of modern analysis, cambridge university press, 1927. How does one motivate the analytic continuation of the riemann zeta function. Analytic continuation of the riemann zeta function 9. Notes on gamma and zeta university of california, berkeley. Analytic continuation of riemann zeta function ijariit. Mod04 lec09 analytic continuation and the gamma function.
The definition of the gamma function occurs as an infinite product weierstrass form. We will say that f can be continued analytically to a region. Mod04 lec10 analytic continuation and the gamma function. This extension arises from repeated application of the recursion relation.
In this chapter, we will see a proof of the analytic continuation of the riemann zeta function. We look at a few of its many interesting properties. Analytic continuation of operators operators acting complex. Finally, we apply our knowledge of analytic continuation to two of the most important functions in analysis, the gamma function and the riemannzeta function, defined originally by a definite. Below, we will present all the fundamental properties of this function, and prove.
Riemanns second proof of the analytic continuation of the riemann zeta function andreas steiger seminar on modular forms, winter term 2006 1 abstract the riemann zetafunction. This then gives and analytic continuation of fs s ss 1 to res 0 with removable singularity at s 1. In this way hypergeometric functions became an important testing ground for riemanns fundamentally new ideas on analytic continuation. We will say that f can be continued analytically to a region d1 that intersects d if there exists a function g, analytic in d1and. Finally, we apply our knowledge of analytic continuation to two of the most important functions in analysis, the gamma function and the riemannzeta function, defined originally by a definite integral and an infinite series, respectively. A maximal analytic continuation of in is an element which continues analytically to the domain, but which cannot analytically be continued to any boundary point of. We shall also develop the idea of analytic continuation. The orange ball marks our current location in the course for an explanation of the background pattern, skip ahead to the end of the page. And since the positive integer n can be chosen arbitrarily, the eulers. Suppose f n is a sequence of functions analytic on an open subset d of c. Complex analysis 7 is analytic at each point of the entire finite plane, then fz is called an entire function. Riemann showed that the function s extends from that halfplane to a meromorphic function on all of c the \riemann zeta function, analytic except for a simple pole at s 1.
Eulers gamma function the gamma function plays an important role in the functional equation for s that we will derive in the next chapter. Mod04 lec10 analytic continuation and the gamma function part ii. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except the nonpositive integers, where the function has simple poles. The gamma function has an analytic continuation to. Notes on the riemann zeta function january 25, 2007 1 the zeta function 1. But along hankels contour c, the integral is welldefined for all z since the contour avoids the origin. On analytic continuation of various multiple zetafunctions kohji matsumoto abstract in this article we describe the development of the problem of analytic continuation of multiple zetafunctions. Felix rubin seminar on modular forms, winter term 2006 abstract. Also, we define a jacobi theta function and xi function with the proof of its functional equation and then it will be used to prove the analytic continuation of zeta function as. Mod04 lec09 analytic continuation and the gamma function part i nptelhrd. We will present proof of analytic continuation of the zeta function using gamma function. The domain of the gamma function extends to negative real numbers by analytic continuation, with simple poles at the negative integers.
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