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2008-05-18T19:13:02Z

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Vipul: New page: ==Statement== The gamma function $\Gamma$ has no zeros. In particular, since we also know that all poles of the gamma function are simple poles, the reciprocal of the g...

2008-05-01T20:01:38Z

New page: ==Statement== The gamma function <math>\Gamma</math> has no zeros. In particular, since we also know that all poles of the gamma function are simple poles, the reciprocal of the g...

New page

==Statement==

The [[gamma function]] <math>\Gamma</math> has no zeros. In particular, since we also know that all poles of the gamma function are [[simple pole]]s, the reciprocal of the gamma function is an [[entire function]].

==Facts used==

* [[Reflection principle for gamma function]]

==Proof==

The reflection principle for the gamma function states that for <math>z</math> not an integer:

<math>\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin (\pi z)}</math>

Thus, if <math>z</math> is not an integer, <math>\Gamma(z) \ne 0</math>. Further, for <math>z</math> a positive integer, <math>\Gamma(z) = (z - 1)!</math>, which is again nonzero. Finally, when <math>z</math> is a negative integer, <math>\Gamma</math> has a simple pole at <math>z</math>.

Gamma function is zero-free - Revision history

Vipul: 1 revision

Vipul: New page: ==Statement== The gamma function \Gamma has no zeros. In particular, since we also know that all poles of the gamma function are simple poles, the reciprocal of the g...

Vipul: New page: ==Statement== The gamma function $\Gamma$ has no zeros. In particular, since we also know that all poles of the gamma function are simple poles, the reciprocal of the g...