How to show lim_(x=> -N)(x+N) Gamma(x)=(-1)^N/N! ?

Cyrus Travis

Cyrus Travis

Answered question

2022-10-30

How to show lim x N ( x + N ) Γ ( x )

Answer & Explanation

Travis Sellers

Travis Sellers

Beginner2022-10-31Added 18 answers

Assuming you are exposed to the Euler's reflection formula:
(1) Γ ( x ) Γ ( 1 x ) = π sin ( π x )
we get:
lim x n ( x + n ) Γ ( x ) = lim x n π ( x + n ) sin ( π x ) 1 Γ ( 1 x ) = y = x + n lim y 0 π y sin ( π y π n ) 1 Γ ( 1 + n y )
Using identity sin ( π y π n ) = ( 1 ) n sin ( π y ), valid for integer n, we conclude
lim x n ( x + n ) Γ ( x ) = ( 1 ) n Γ ( 1 + n ) lim y 0 π y sin ( π y ) lim z 0 z sin ( z ) = 1 = ( 1 ) n n !

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