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Legendre Duplication Formula

المؤلف:  Abramowitz, M. anHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Doverd Stegun, I. A.

المصدر:  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover

الجزء والصفحة:  ...

22-5-2019

5786

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)
			(4)
			(5)
			(6)

Now, use the beta function identity

(7)

to write the above as

(8)

Solving for  and using  then gives

			(9)
			(10)

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REFERENCES:

Abramowitz, M. and Stegun, I. 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.

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 5, 1981.

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