Integration and Differentiation of Factorials.

I was solving some problems regarding the Gamma Function the other day, and wondered if it would be possible to integrate or differentiate x! with respect to x. This would be $\int x! dx$ or $\dfrac{\text{d}}{\text{d}x} x!$ .

I've been trying to solve do this through the use of the Pi Function and integrating it, which would be $\int Γ(x+1) dx$ or $\displaystyle\int\displaystyle\int_{0}^{\infty} (z^ {x} e^{-z}) dz dx$ . I wasn't able to solve this, and wanted to know if it was possible to do this, and if so, what would the integral/derivative of x! with respect to x be?

#Calculus

Easy Math Editor

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Comments

See Digamma function.

Pi Han Goh - 2 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$