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Chapter 4: Problem 14

Die Gauss'sche e-Funktion sei definiert durch \(\psi(z):=\Gamma^{\prime}(z) / \Gamma(z)\). Man seige: a) \(v\) ist meromorph in \(\mathbb{C}\) mit einfachen Polstellen in \(S=\left\\{-n ; \quad n \in N_{0}\right\\}\) und \(\operatorname{Res}(\psi ;-n)=-1\) b) \(\psi(1)=-\gamma\) (EULER-MASCHERONI'sche Konstante). c) \(\psi(z+1)-\psi(z)=\frac{1}{z}\). d) \(\psi(1-z)-\psi(z)=\pi \cot \pi z\). e) \(\psi(z)=-\gamma-\frac{1}{z}-\sum_{v=1}^{\infty}\left(\frac{1}{z+\nu}-\frac{1}{\nu}\right)\) f) \(\psi^{\prime}(z)=\sum_{\nu=0}^{\infty} \frac{1}{(z+\nu)^{2}}\), wobei die Reihe rechts in \(\mathrm{C}\) normal konvergiert. g) Far positives \(x\) ist $$ (\log T)^{n}(x)=\sum_{v=0}^{\infty} \frac{1}{(x+\nu)^{2}}>0 $$ die reelle \(I\)-Funktion ist also logarithmisch konvex.

Short Answer

Expert verified
The digamma function \( \psi(z) \) satisfies given properties involving meromorphic properties, special values, relations and series expansions.

Step by step solution

01

Understanding the Function

The Gauss's digamma function is defined as \( \psi(z) = \frac{\Gamma'(z)}{\Gamma(z)} \), where \( \Gamma(z) \) is the gamma function. This function is important in many areas of mathematics, especially in analytic number theory and complex analysis.
02

Analyzing Meromorphic Properties

To show that \( \psi(z) \) is meromorphic with simple poles at \( S = \{ -n; \quad n \in N_0 \} \) and \( \operatorname{Res}(\psi ;-n)=-1 \), we begin by understanding that \( \Gamma(z) \) has simple poles at these points. The derivative \( \Gamma'(z) \) does not add any new poles, so \( \psi(z) \) retains these poles. The residue at any pole \(-n\) can be shown to be \(-1\) by observing the behavior of \( \Gamma(z) \) around \( z = -n \).
03

Evaluating Special Values

For \( \psi(1) = -\gamma \), we consider special cases of \( \psi(z) \). The value at \( z = 1 \) directly gives the negative of the Euler-Mascheroni constant \( \gamma \). This is derived from properties of the gamma function and its derivatives evaluated at 1.
04

Functional Equation Verification

To prove \( \psi(z+1)-\psi(z)=\frac{1}{z} \), use the recurrence relation of the \( \Gamma \) function, which states \( \Gamma(z+1) = z \Gamma(z) \). Differentiating and applying the definition of the digamma function, we obtain the discrete difference equation as a consequence of the logarithmic derivative property.
05

Reflection Formula Derivation

For \( \psi(1-z)-\psi(z)=\pi \cot \pi z \), use Euler's reflection formula for the gamma function that relates \( \Gamma(z) \) and \( \Gamma(1-z) \). The digamma function's difference expresses the periodicity and symmetry properties via trigonometric identities involving cotangent.
06

Series Expansion

The formula \( \psi(z)=-\gamma-\frac{1}{z}-\sum_{v=1}^{\infty}\left(\frac{1}{z+u}-\frac{1}{u}\right) \) can be derived from the digamma function's known expansions, where the series converges, giving the structure of \( \psi \). This expansion helps understand the behavior at large \( z \) and near pole locations.
07

Derivative Representation

To show \( \psi^{\prime}(z)=\sum_{u=0}^{\infty} \frac{1}{(z+u)^{2}} \), differentiate the series representation of \( \psi(z) \) term by term. The series is uniformly convergent in \( \mathbb{C} \), allowing term-wise differentiation that results in this square term series.
08

Logarithmic Convexity of Series

For large positive \( x \), \( \log(T)^n(x) = \sum_{v=0}^{\infty} \frac{1}{(x+u)^2} > 0 \). This assures that the real-valued function is logarithmically convex by showing each term in the series is positive, leading to the entire series being positive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gamma function
The Gamma function, denoted as \( \Gamma(z) \), is a cornerstone in various areas of mathematics, especially in analytic number theory and complex analysis. It generalizes the factorial function to complex numbers. For any positive integer \( n \), the Gamma function is defined as \( \Gamma(n) = (n-1)! \). But more remarkably, it is extended to non-integer and complex values using the integral formula:
  • \( \Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t} dt \) for \( \text{Re}(z) > 0 \).

The Gamma function is particularly useful because it exhibits a recursion property similar to factorials: \( \Gamma(z+1) = z\Gamma(z) \). This property allows for significant flexibility in mathematical analysis and is essential for proving many identities and equations.
Additionally, it aids in defining other related functions like the digamma function \( \psi(z) = \frac{\Gamma'(z)}{\Gamma(z)} \), crucial in analyzing special values and relationships of functions.
meromorphic functions
In the context of complex analysis, a meromorphic function is one that is holomorphic (or complex differentiable) throughout its domain except for isolated poles, which are points where the function's value grows to infinity. The digamma function \( \psi(z) \) is one such meromorphic function. It inherits these properties from the Gamma function \( \Gamma(z) \), which has simple poles at non-positive integers \( -n \).
  • Meromorphic functions combine features of both holomorphic functions and functions with isolated singularities.
  • They are crucial in mathematical studies that explore the behavior of complex functions, zeros, and singularities.
This property of having poles means the digamma function can be analyzed using residues: for example, the residue at a simple pole \( -n \) is \(-1\), a direct result of how \( \Gamma(z) \) behaves near its poles. These properties are instrumental in both theoretical and applied mathematics.
Euler-Mascheroni constant
The Euler-Mascheroni constant, represented by \( \gamma \), is a fascinating number that arises in number theory and analysis. It is approximately 0.57721 and occurs in various contexts, such as in the harmonic series and integrals. Its exact nature remains a mystery, as it is not known if \( \gamma \) is rational or irrational.
  • \( \gamma \) surfaces in limits and approximations, manifesting in expressions like \( \psi(1) = -\gamma \).
  • This negative sign comes from evaluating the digamma function at 1, utilizing the properties of the Gamma function.
The constant shows up in the series expansions and particular values of the Gamma and digamma functions. The role of \( \gamma \) in these functions helps to simplify complex mathematical expressions and leads to significant insights into the behavior of logarithmic and harmonic functions.
analytic number theory
Analytic number theory is a branch focused on the use of analysis techniques to solve problems about integers. This field leverages tools from complex analysis, including functions like the Gamma and digamma functions, because these functions have properties that align well with number-theoretic operations.
  • The Gamma function exhibits periodicity, factorial generalization, and connects to the Riemann zeta function, a pivotal element in analytic number theory.
  • Digamma function鈥檚 properties help simplify certain mathematical proofs and relationships within number theory.
Analytic number theory examines properties of numbers through their functions and aids in understanding distribution of prime numbers, divisors, and other number properties. Techniques like residue calculus, functions' differential properties, and series expansions, rooted in this theory, lead to abundant results about numbers and their fundamental nature.

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Most popular questions from this chapter

Man bestimme eine in \(\mathbb{C}\) meromorphe Funktion \(f\), die in der Menge $$ S=\\{\sqrt{n} ; \quad n \in \mathrm{N}\\} $$ cinfache Polstellen mit \(\operatorname{Res}(f ; \sqrt{n})=\sqrt{n}\) hat und in \(C-S\) analytisch ist.

Man zeige: $$ \Gamma\left(\frac{1}{6}\right)=2^{-1 / 3}\left(\frac{3}{\pi}\right)^{1 / 2} \Gamma\left(\frac{1}{3}\right)^{2} $$

Man zeige, da脽 die durch $$ \gamma_{n}:=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}-\log n $$ definierte Folge \(\left(\gamma_{n}\right)\) (streng) monoton fallt und durch 0 nach unten beschr盲nkt ist. Also existiert $$ \gamma:=\lim _{n \rightarrow \infty} \gamma_{n}=0,577215664901532860606.512090082402431042159 \ldots $$ (Eiler-Mascheroni'sche Konstante).

F眉r \(z \in \mathrm{C}-S\) gilt $$ \lim _{n \rightarrow \infty} \frac{\Gamma(z+n)}{n^{2} \Gamma(n)}=1 $$ dabei sei \(S=\\{0,-1,-2,-3, \ldots\\}\)

Sei \(f: \mathbb{C} \rightarrow \bar{C}\) eine meromorphe Funktion mit lauter einfachen Polen und ganzzahligen Residuen. Dann gibt es eine meromorphe Funktion \(h: \mathrm{C} \longrightarrow\) \(\overline{\mathrm{C}}\) mit \(f(z)=h^{\prime}(z) / h(z)\).
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