International J. of Pure & Engg. Mathematics (IJPEM) ISSN 2348-3881, Vol. 5 No. III (December, 2017), pp. 39-43

A NOTE ON BETA FUNCTION AND FOURIER TRANSFORM

P. B. GIRASE1 AND V. C. BORKAR2 1 Department of Mathematics, K. K. M. College, Manwath, Dist:Parbhani(M.S.)-431505, India 2 Department of Mathematics, Yeshwant Mahavidyalaya, Nanded (M.S.)-431602, India

Abstract In this paper, we use the Fourier Transform and convolution theorem of Fourier Transform to prove the relation between Gamma function and beta function. It is possible that this technique of proof may be applied to solve the other problems involving beta function.

1. Introduction The beta function [1], also called the Euler integral of the first kind, is a special function R 1 p−1 q−1 defined by B(p, q) = 0 x (1 − x) dx ,for Re(p) > 0, Re(q) > 0. The gamma function [2], also called the Euler integral of the second kind, is defined as ∞ convergent improper integral Γ (n) = R e−xxn−1dx for Re(n) > 0. 0 Properties of beta function:

−−−−−−−−−−−−−−−−−−−−−−−−−−−− Key Words : Beta function, Gamma function, Fourier transform, Convolution Theorem. 2010 AMS Subject Classification : 33B15, 42A38, 42B10. c http: //www.ascent-journals.com

39 40 P. B. GIRASE & V. C. BORKAR

1. B(p, q) = B(q, p)

R ∞ xp−1 2. B(p, q) = 0 (1+x)p+q dx, Re(p) > 0, Re(q) > 0

π R 2 2p−1 2q−1 3. B(p, q) = 2 0 sinθ cosθ dx, Re(p) > 0, Re(q) > 0

Γ(p)Γ(q) The key property of beta function is B(p, q) = Γ(p+q) ,which is to be proved in main results by using convolution theorem of Fourier transform. Identities of beta and gamma function:

1. B(p, q) = B(p + 1, q) + B(p, q + 1), p, q > 0

q q 2. B(p, q + 1) = p B(p + 1, q) = p+q B(p, q), p, q > 0

3. Γ (p + 1) = pΓ(p), p > 0

The proof of the relation between Gamma function and Beta function requires the following definitions and results. Definition 1.1 [5] : The Fourier transform of the complex valued function f(x) is R ∞ isx defined by the integral, F [f(x)] = F (s) = −∞ f(x)e dx, provided that integral exists for complex parameter s, and the Inverse Fourier transform of F (s) is defined by the 1 R ∞ −isx integral f(x) = 2π −∞ F (s)e ds. Definition 1.2 [4] : Convolution: The convolution of two functions f(x) and g(x) over R ∞ the interval (−∞, ∞) is defined as f(x) ∗ g(x) = −∞ f(u)g(x − u)du Lemma 1.3 [5] : Convolution Theorem : The Fourier transform of the convolution of f(x) and g(x) is the product of their Fourier transforms, that is F [f(x) ∗ g(x)] = R ∞ F [ −∞ f(u)g(x − u)du] = F [f(x)]F [g(x)]. Definition 1.4 [5] : Heaviside’s unit function: The Heaviside’s unit step function u(t- a) is defined as follows:

 0 t < a u(t − a) = (1) 1 t > a The product

 0 t < a f(t)u(t − a) = (2) f(t) t > a A NOTE ON BETA FUNCTION AND FOURIER TRANSFORM... 41

2. Main Results Theorem 2.1 : Relation between Gamma function and Fourier transform:If s > 0, n−1 n Γ(n) then F [u(t)t ] = i sn , where u(t) is the unit step function and n > 0. Proof : Z ∞ Consider, F [u(t)tn−1] = u(t)tn−1eistdt (3) −∞ Z 0 Z ∞ = u(t)tn−1eistdt + u(t)tn−1eistdt (4) −∞ 0 Z ∞ = tn−1eistdt (5) 0 −x −dx put ist = −x, t = is , dt = is

Z ∞ −x −dx F [u(t)tn−1] = ( )n−1e−x( ) (6) 0 is is Z ∞ n 1 −x n−1 = i n e x dx (7) s 0 Γ(n) = in (8) sn 2 R 1 m−1 n−1 Γ(m)Γ(n) Theorem 2.2 : If m > 0, n > 0, then B(m, n) = 0 x (1 − x) dx = Γ(m+n) . m−1 m Γ(m) n−1 n Γ(n) Proof : Let F [u(t)t ] = i sm , F [u(t)t ] = i sn We have, by convolution theorem, Z ∞ F [f(t)]F [g(t)] = F [f(t) ∗ g(t)] = F [ f(y)g(t − y)dy] (9) −∞ therefore Z ∞ F [u(t)tm−1]F [u(t)tn−1] = F [ u(y)ym−1u(t − y)(t − y)n−1dy] (10) −∞ that is, Z ∞ m Γ(m) n Γ(n) m−1 n−1 i m i n = F [ u(y)y u(t − y)(t − y) dy] (11) s s −∞ R ∞ m−1 n−1 Let h(t) = −∞ u(y)y u(t − y)(t − y) dy Since,  1 0 < y < t, t 0 u(y)u(t − y) = > [3] (12) 0 Otherwise 42 P. B. GIRASE & V. C. BORKAR

Z t h(t) = ym−1(t − y)n−1dy (13) 0 that is,

 R t ym−1(t − y)n−1dy t 0 h(t) = 0 > (14) 0 t < 0 or Z t h(t) = u(t) ym−1(t − y)n−1dy, [3] (15) 0 where

 1 t 0 u(t) = > (16) 0 t < 0 therefore , equation (11) becomes,

Z t m+n Γ(m)Γ(n) m−1 n−1 i m+n = F [u(t) y (t − y) dy] (17) s 0 put, y = tx, dy = tdx, (new limits are x=0, x=1),

Z 1 m+n Γ(m)Γ(n) m−1 n−1 i m+n = F [u(t) (tx) (t − tx) tdx] (18) s 0 Z 1 = F [u(t)tm+n−1] xm−1(1 − x)n−1dx (19) 0 Γ(m + n) = im+n B(m, n) (20) sm+n Therefore, Γ(m)Γ(n) = B(m, n) (21) Γ(m + n) 2 3. Conclusion The Fourier transform technique can be used to solve problems involving Beta function instead of probability theory. A NOTE ON BETA FUNCTION AND FOURIER TRANSFORM... 43

References

[1] urlhttp://en.wikipedia.org/wiki/Beta function [2] http://en.wikipedia.org/wiki/Gamma function [3] courses.washington.edu/bioen316/Assignments/316 SCP.pdf [4] https://en.wikipedia.org/wiki/Convolution#Definition [5] Grewal B. S. and Grewal J. S., Higher Engineering Mathematics,40th Edition, Khanna Publishers, New Delhi, (2007).