Gamma function

  • #1
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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  • #2
gleem
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Try multiplying the numerator and denominator by z and pull out an n from the exponent then rearrange terms.
 
  • #3
I'm asking about the final step, How to get the final step from the second step?
 
  • #4
gleem
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The limit of the product is product of the limit of the factors. What is the definition of the gamma function?
 
  • #5
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?
 
  • #6
HallsofIvy
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So you are asking specifically about
[tex]\lim_{n\to \infty} \frac{nz}{z+ n+ 1}[/tex]

Divide both numerator and denominator by n:
[tex]\lim_{n\to\infty}\frac{z}{\frac{z}{n}+ 1+ \frac{1}{n}}[/tex]
 
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  • #7
gleem
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nz/(n+z+1) → nz/n in the limit n→∞
 
  • #8
So you are asking specifically about
[tex]\lim_{n\to \infty} \frac{nz}{z+ n+ 1}[/tex]

Divide both numerator and denominator by n:
[tex]\lim_{n\to\infty}\frac{z}{\frac{z}{n}+ 1+ \frac{1}{n}}[/tex]

Yeah that's what I asked for, Thank you!
 

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