Derivative of the Gamma Function

In summary, the derivative of the gamma function can be expressed as \frac{d}{dz} \Gamma(z) = \Gamma(z) \cdot \psi^{(m)}(z), where \Gamma(z) is the gamma function and \psi^{(m)}(z) is the polygamma function. This can also be written as \frac{d}{dz} \Gamma(z) = \left( \int_0^\infty t^{z-1} e^{-t} \, dt \right) \left( \int_0^{\infty}\left(\frac{e^{-t}}{t} - \frac{e^{-zt}}{1 - e^{-t}}
  • #1
Unit
182
0
A very vague question:

What is the derivative of the gamma function?
Here's what I've got, using differentiation under the integral. Can anybody tell me if I'm on the right track? What does my answer mean?

[tex]\Gamma(z) = \int_0^{\infty} t^{z - 1} \: e^{-t} \; dt[/tex]

The integrand can be expressed as a function [itex]f(z, t) = t^{z - 1} \: e^{-t}[/itex]:

[tex]\frac{d}{dz} \int_{y_0}^{y_1} f(z, t) \; dt = \int_{y_0}^{y_1} \frac{\partial}{\partial z} f(z, t) \; dt [/tex]

Because [itex]e^{-t}[/itex] is a constant of sorts (correct me if I'm wrong), with respect to the partial derivative to z, all this turns into

[tex]\frac{d}{dz} \Gamma(z) = \int_0^{\infty} t^{z-1} \: e^{-t} \: \ln(t) \; dt[/tex]

But this is useless! :yuck: Does anybody have any thoughts?

(I'm trying to find all the extrema of the gamma function, ... they look like the follow an exponential curve and I want to see if there is an expression for it)
 
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  • #2
Unit said:
[tex]\frac{d}{dz} \Gamma(z) = \int_0^{\infty} t^{z-1} \: e^{-t} \: \ln(t) \; dt[/tex]

But this is useless! :yuck:

nevertheless correct. [tex]\Gamma'[/tex] cannot be written in simpler ways. You often see [tex]\Psi(x) = \Gamma'(x)/\Gamma(x)[/tex] called the digamma function.

(I'm trying to find all the extrema of the gamma function, ... they look like the follow an exponential curve and I want to see if there is an expression for it)

These extrema are for negative x, right? But the integral form converges only for positive x, so it is of no use in this.
 
  • #3
Yeah, although it would be cool if THIS non-analytical function could have an analytical derivative... alas... :'(
 
  • #4
wait, n1person, are you hinting that there exist OTHER non-analytical functions that could have an analytical derivative?

p.s. thank you both for your replies :smile:
 
  • #5
Derivative of the Gamma Function...


Unit said:
What is the derivative of the gamma function?

Derivative to the gamma function according to Mathematica 6:
[tex]\frac{d}{dz} \Gamma(z) = \Gamma(z) \cdot \psi^{(m)}(z)} = \left( \int_0^\infty t^{z-1} e^{-t} \, dt \right) \left( \int_0^{\infty}\left(\frac{e^{-t}}{t} - \frac{e^{-zt}}{1 - e^{-t}}\right) \, dt \right) \; \; \; m = 0[/tex]

[tex]\boxed{\frac{d}{dz} \Gamma(z) = \left( \int_0^\infty t^{z-1} e^{-t} \, dt \right) \left( \int_0^{\infty}\left(\frac{e^{-t}}{t} - \frac{e^{-zt}}{1 - e^{-t}}\right) \, dt \right) \; \; \; m = 0}[/tex]

[tex]\Gamma(z)[/tex] - Gamma function
[tex]\psi^{(m)}(z)[/tex] - Polygamma function
The m^th derivative of the digamma function.

Reference:
http://en.wikipedia.org/wiki/Gamma_function" [Broken]
http://en.wikipedia.org/wiki/Digamma_function" [Broken]
http://en.wikipedia.org/wiki/Polygamma_function" [Broken]
 
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  • #6
Well any function that is used to define an anti-derivative/integral of an analytical function would have an analytical derivative, right?
[tex]
\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt.
[/tex]

so the derivative of error function;

[tex]
\frac{\rm d}{{\rm d}x}\,\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\,e^{-x^2}.
[/tex]
 

1. What is the derivative of the gamma function?

The derivative of the gamma function is a mathematical function that represents the rate of change of the gamma function with respect to its input. It is denoted by Γ'(x) or dΓ(x)/dx.

2. How is the derivative of the gamma function calculated?

The derivative of the gamma function can be calculated using the formula Γ'(x) = Γ(x) * (Ψ(x) - ln(x)), where Γ(x) is the gamma function and Ψ(x) is the digamma function.

3. What is the relationship between the gamma function and the derivative of the gamma function?

The gamma function and its derivative are closely related as they both involve the use of the digamma function. The gamma function is the integral of the derivative of the gamma function, and the derivative of the gamma function can be expressed in terms of the gamma function itself.

4. Why is the derivative of the gamma function important in mathematics?

The derivative of the gamma function is an important tool in mathematics as it is used to solve problems in many different fields, including statistics, physics, and engineering. It is also a key component in the development of many mathematical models and functions.

5. Can the derivative of the gamma function be extended to complex numbers?

Yes, the derivative of the gamma function can be extended to complex numbers using the Cauchy-Riemann equations. This extension is known as the complex derivative of the gamma function and is denoted by Γ'(z) or dΓ(z)/dz.

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