# Gamma function

This is given in Mathematical methods for physicists by Arfken and weber, while defining a property of gamma function, I have no idea how the term in the red circle becomes 'z' in the final step, please help

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gleem
Try multiplying the numerator and denominator by z and pull out an n from the exponent then rearrange terms.

I'm asking about the final step, How to get the final step from the second step?

gleem
The limit of the product is product of the limit of the factors. What is the definition of the gamma function?

I know the limit of the second term in the product converges to "gamma of z', it is how the gamma function is defined, But I couldn't figure out how the limit of the first term in the product converges to 'z'. How do I compute it?

HallsofIvy
Homework Helper
$$\lim_{n\to \infty} \frac{nz}{z+ n+ 1}$$

Divide both numerator and denominator by n:
$$\lim_{n\to\infty}\frac{z}{\frac{z}{n}+ 1+ \frac{1}{n}}$$

JR Sauerland and Muthumanimaran
gleem
nz/(n+z+1) → nz/n in the limit n→∞

$$\lim_{n\to \infty} \frac{nz}{z+ n+ 1}$$
$$\lim_{n\to\infty}\frac{z}{\frac{z}{n}+ 1+ \frac{1}{n}}$$