TOPICS
Search

Legendre Duplication Formula

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

Gamma functions of argument 2z can be expressed in terms of gamma functions of smaller arguments. From the definition of the beta function,

B(m,n)=(Gamma(m)Gamma(n))/(Gamma(m+n))=int_0^1u^(m-1)(1-u)^(n-1)du.
(1)

Now, let m=n=z, then

(Gamma(z)Gamma(z))/(Gamma(2z))=int_0^1u^(z-1)(1-u)^(z-1)du
(2)

and u=(1+x)/2, so du=dx/2 and

(Gamma(z)Gamma(z))/(Gamma(2z))=int_(-1)^1((1+x)/2)^(z-1)(1-(1+x)/2)^(z-1)(1/2dx)
(3)
=1/2int_(-1)^1((1+x)/2)^(z-1)((1-x)/2)^(z-1)dx
(4)
=1/(2^(1+2(z-1)))int_(-1)^1(1-x^2)^(z-1)dx
(5)
=2^(1-2z)[2int_0^1(1-x^2)^(z-1)dx].
(6)

Now, use the beta function identity

B(m,n)=2int_0^1x^(2m-1)(1-x^2)^(n-1)dx
(7)

to write the above as

(Gamma(z)Gamma(z))/(Gamma(2z))=2^(1-2z)B(1/2,z)=2^(1-2z)(Gamma(1/2)Gamma(z))/(Gamma(z+1/2)).
(8)

Solving for Gamma(2z) and using Gamma(1/2)=sqrt(pi) then gives

Gamma(2z)=(2pi)^(-1/2)2^(2z-1/2)Gamma(z)Gamma(z+1/2)
(9)
=(2^(2z-1)Gamma(z)Gamma(z+1/2))/(sqrt(pi)).
(10)

See also

Gamma Function, Gauss Multiplication Formula

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Legendre Duplication Formula

Cite this as:

Weisstein, Eric W. "Legendre Duplication Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LegendreDuplicationFormula.html

Subject classifications