“Legendre Duplication Formula”的概念、定义、习题解题方法及翻译-数学辞典

单词

Legendre Duplication Formula

释义

Legendre Duplication Formula

Gamma Functions of argument can be expressed in terms of Gamma Functions of smaller arguments. From the definition of the Beta Function,

(1)

Now, let

, then

(2)

and

, so

and




			(3)

Now, use the Beta Function identity

(4)

to write the above as

(5)

Solving for


			(6)

since

See also Gamma Function, Gauss Multiplication Formula

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 256, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 561-562, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 424-425, 1953.

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