Derivative of the Gamma Function

• loto
In summary, the derivative of the Gamma Function is an important function in mathematics that is used in various areas such as complex analysis, number theory, and statistics. Its formula is given by d/dx(Gamma(x)) = Gamma(x) * (Psi(x) - ln(x)), and it can be simplified using different properties of the Gamma Function. In probability and statistics, the derivative of the Gamma Function is used to calculate the probability density function and cumulative distribution function of the Gamma distribution, a commonly used model for continuous random variables.
loto
Hi gang,

I'm having trouble with doing a derivative of the Gamma function. I know both the definition of Gamma and Polygamma, but can't see how to get from the derivative of Gamma to Psi times Gamma. Any help or hints would be great.

Thanks!

But isn't it the very definition of $\psi^{(0)}$ that $\Gamma'(z)=\psi^{(0)}\Gamma(z)$?

I think you just have to differentiate $\Gamma(z)$. Since everything is continuous, you can move the derivative inside the integral, etc.

1. What is the derivative of the Gamma Function?

The derivative of the Gamma Function is the function obtained by differentiating the Gamma Function with respect to its argument.

2. Why is the derivative of the Gamma Function important?

The derivative of the Gamma Function is important in many areas of mathematics, including complex analysis, number theory, and statistics. It is also used in the calculation of many integrals involving the Gamma Function.

3. What is the formula for the derivative of the Gamma Function?

The formula for the derivative of the Gamma Function is given by: d/dx(Gamma(x)) = Gamma(x) * (Psi(x) - ln(x)), where Psi(x) is the digamma function.

4. Can the derivative of the Gamma Function be simplified?

Yes, the derivative of the Gamma Function can be simplified using various properties of the Gamma Function, such as its relationship with the Beta Function and the recurrence relation for the Gamma Function.

5. How is the derivative of the Gamma Function used in probability and statistics?

The derivative of the Gamma Function is used in probability and statistics to calculate the probability density function and cumulative distribution function of the Gamma distribution, which is commonly used to model continuous random variables.

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