7E (Further Complex Methods)

For

Re z > 0

I (z + 1) = \int_{0}^{\infty} t^{z} e^{- t} d t = [- t^{z} e^{- t}]_{0}^{\infty} + z \int_{0}^{\infty} t^{z - 1} e^{- t} d t .

I (z + 1) = z I (z) .

ii

Given any

z \in C

that is not a nonpositive integer, choose

N

large enough that

Re (z + N) > 0

I (z) = \frac{I ( z + N )}{z ( z + 1 ) \dots ( z + N - 1 )} .

The recurrence shows that definitions from different sufficiently large

N

agree, so this gives the unique analytic continuation from the right half-plane. It is analytic except where a denominator factor vanishes, namely

z = 0, - 1, - 2, \dots .

These points are simple poles. More precisely, at

z = - n

Res_{z = - n} I (z) = \frac{( - 1 ) ^{n}}{n !},

because one may write

I (z) = \frac{I ( z + n + 1 )}{z ( z + 1 ) \dots ( z + n )}

and use

I (1) = 1

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