# What is the integral representation of the Digamma function?

• MHB
• DreamWeaver
In summary, the Gamma function is defined as the integral of t raised to the power of x-1, multiplied by e to the power of -t, from 0 to infinity. The Digamma function is the derivative of the natural logarithm of the Gamma function, or the ratio of the derivative of Gamma to Gamma itself. Dirichlet's integral representation for the Digamma function states that it can also be expressed as an integral of a complex function involving e to the power of -z and (1+z) raised to the power of -x, from 0 to infinity.
DreamWeaver
For the Gamma function:

$$\displaystyle \Gamma(x) = \int_0^{\infty}t^{x-1}e^{-t}\, dt$$And the Digamma function:

$$\displaystyle \psi_0(x) = \frac{d}{dx}\log \Gamma(x) = \frac{\Gamma'(x)}{\Gamma(x)}$$Prove Dirichlet's integral representation for the Digamma function:$$\displaystyle \psi_0(x) = \int_0^{\infty} \frac{1}{z}\left( e^{-z} - \frac{1}{(1+z)^x} \right)\, dz$$Hint:

Evaluate the double integral

$$\displaystyle \int_{0}^{\infty}\int_{1}^{q}e^{-tz}\, dt\, dz$$

in two different ways, and equate the results.

Consider

$$f(t) = \int^\infty_0 z^{t-1}\left(e^{-z}-\frac{1}{(1+z)^x} \right)dz = \Gamma(t)-\frac{ \Gamma(t) \Gamma(x-t)}{ \Gamma(x)}$$

$$f(t)= \frac{\Gamma(1+t)}{\Gamma(x)}\frac{ \Gamma(x) -\Gamma(x-t)}{t}$$

Now take the limit as $t\to 0$

$$\frac{1}{\Gamma(x)}\lim_{t \to 0}\frac{ \Gamma(x) -\Gamma(x-t)}{t}= \frac{\Gamma'(x)}{\Gamma(x)}=\psi(x)$$

ZaidAlyafey said:
Consider

$$f(t) = \int^\infty_0 z^{t-1}\left(e^{-z}-\frac{1}{(1+z)^x} \right)dz = \Gamma(t)-\frac{ \Gamma(t) \Gamma(x-t)}{ \Gamma(x)}$$

$$f(t)= \frac{\Gamma(1+t)}{\Gamma(x)}\frac{ \Gamma(x) -\Gamma(x-t)}{t}$$

Now take the limit as $t\to 0$

$$\frac{1}{\Gamma(x)}\lim_{t \to 0}\frac{ \Gamma(x) -\Gamma(x-t)}{t}= \frac{\Gamma'(x)}{\Gamma(x)}=\psi(x)$$
Crikey! That was a very quick proof... Very impressive! (Rock)(Rock)(Rock)

Not that quick. Especially if I had used my phone. Don't post lots of interesting questions for otherwise I'll spend the whole day typing... Just kiddin'

ZaidAlyafey said:
Not that quick. Especially if I had used my phone. Don't post lots of interesting questions for otherwise I'll spend the whole day typing... Just kiddin'

He he! Just for that, I'm going to post more, not less... (Hug)

## 1. What is Dirichlet's integral?

Dirichlet's integral, also known as the Dirichlet's eta function, is a mathematical function that represents the alternating harmonic series. It is named after German mathematician Peter Gustav Lejeune Dirichlet.

## 2. What is the formula for Dirichlet's integral?

The formula for Dirichlet's integral is ζ(s) = ∑n=1∞ (-1)n-1/ns, where ζ is the Riemann zeta function and s is the input variable. It is a special case of the more general Dirichlet series.

## 3. What are the applications of Dirichlet's integral?

Dirichlet's integral has applications in number theory, complex analysis, and physics. It is used to study the distribution of prime numbers, to define the Riemann zeta function, and to solve certain differential equations. It also plays a role in the study of the Riemann hypothesis.

## 4. How is Dirichlet's integral related to the Basel problem?

The Basel problem, also known as the Basel sum, asks for the exact value of the infinite sum ∑n=1∞ 1/n2. This problem was solved by Leonhard Euler using the technique of analytic continuation, which is closely related to Dirichlet's integral. In fact, Euler's solution can be generalized to find the exact value of ∑n=1∞ 1/ns for any s greater than 1.

## 5. Are there any open problems related to Dirichlet's integral?

Yes, there are several open problems related to Dirichlet's integral. One of the most famous is the Riemann hypothesis, which states that all nontrivial zeros of the Riemann zeta function have a real part of 1/2. Another open problem is the generalized Riemann hypothesis, which extends the Riemann hypothesis to Dirichlet L-functions. Both of these problems have connections to Dirichlet's integral and are considered major unsolved problems in mathematics.

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