The Gamma Function (Dover Books on Mathematics) by Emil Artin

Author:Emil Artin [Artin, Emil]
Language: eng
Format: azw3
Publisher: Dover Publications
Published: 2015-01-28T05:00:00+00:00

The general term is smaller than 1/i2 because x > 0. This series obviously converges uniformly for positive x. Repeated differentiation leads to ever better converging series, all of which converge uniformly for positive x. This shows that log Γ(x) can be differentiated as often as we please. We have the formula

There is no trouble in extending the validity of Eq. (2.11) to include negative x. This is easily done with the help of Eq. (2.2). We merely determine the functional equation that the left side of Eq. (2.11) satisfies, and then show that the right side also satisfies it. Both these steps are quite obvious.

The case k = 2 is of special interest. The function

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