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4), , Beta and Gamma, , 4-1. Gamma Functions, We define Gamma functions, , T(n) (read Gamma n) by the integral, -1dt, n>0 (Parvanchal 2012, MGKV 2014, , T()=, These, , Functions, , as, , also known as Eulerian Integrals of second kind after the, Mathematician Euler who introduced them in 1729., S 4-2. An, , are, , name of, , Important Property., , To prove, , that T(n +, , We have r, , 1), , =nTnsor all values of n., , (Gorakhpur 83), , (n)= J e-1dt, , Integrating by parts taking "as first function, we have, , r, , 0, , -, , -1)P-2(-e), en-2 dt, , 0+(n1), , (n -1), =, , (n -1) T(n, , -2d, -, , 1), , replacing n by (n + 1), we get, T(n+1)nt (n) as required., Cor I., , by definition, , ro=1, r(1) ef-1de, , r, a, r(1)=1, , for, , Cor II. If n is a, We know that r positive, +, , positive, , integralT (n +1), 1) =n (n} for alta>=n, 0, applying this resu, integer n, we have, T(n +1) = n (n 1) (n -2), n l as, T(1) 1. .3.2.1r (1), (n, , -, , =, , S 4:3., , Transformation of Gamma Functions., , We have rn), , =, , Jen-1dt, , success

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100, , Integrals Calculus, T(/2)=2, , -2, Multiplying (1) and (2), , ...(1), , d, Property, , we get, , 1 of, , 2), , definite integrals, , edy, , -4, , On transposing to polars,, , ertthedy, we, , get, , t/2, , e i de dr, , T, , from article 4-4., , de =2, , d, , t/2, 2(, (1/2) VT., =, , Remark. If n is, positive integer than the, v t gives the formula, , formula, , T, , (m, , +, , 1), , =, , mT, , (m), , and, , --, , In particular, , -, , S 4-6. Beta functions., Beta, the integral, , functions, , denoted by, , are, , also known, , (m,n),, , read, , we, , Proof., , (MGKV 2014), , the Eulerian, integral of first kind., of Beta functions., , shall prove the B (m,, n), , B(m, n) J, B, , (m, n), , we, , =, , B, , (n, m)., , -1(1-xy-1 de from definition., , =, , Putting x =1-y,, , m,n > 00, , (Gorakhpur 80), , get, , =, , (1-yy1y"-1dy, -1(1 - yy -dy, , S, , Beta m,n and is defined, by, , as, , as, , 4 7. Symmetry property, Here, , B, , Bm, n)= J-11-y-Id, , These, , S, , are, , 4-8. An useful transformation., , B (m, n) = J, , =, , B, , (n, m)., , Proved, , (Gorakhpur 81-, , -(1--14, .(1

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101, /, , wna Functions, , int, , 1, , s o that, , 1+ty, , 1r14, , *, , (1+, We get, , -J (, , B (m, m)-, , -1, , dy, (1 +y), , y1dy, , -, , (1 +yyn +n, , ..2), , 11., . Since B (n, n) =B (1, m) hence by interchanging m and n in (2),, Remark, , B (m, n), , yn-1 dy, , =, , (1 +yym +n, , 3), , aark 2. Relation (2) or (3) provide alternative definitions of Beta functions., nar:, , Relation between Beta and Gamma functions., b prove, , that B, , (m, n), , =, , m)Ta), , r(m +n), , (Gorakhpur 87, 98, 2003, 10; Purv. 96 98, 2012; MGKV 2013, 14), We know that, , T)- J e"p-1dt, , .1), , where z is a constant, , T(n) = J, , e, , -1dt, , Multiplying both sides by e72" ,, Tn)e-1=, , we get, , zn +m, , 1) e - z ( + 1) m - 1dt, , Now integrating both sides with respect to z within the limits 0 and co and, , the definition, , 1dz, , =r(m), we havee, e-z( +1) (n +n 1) dzl- 1 dt, , Tn)r(m) =, But, , -, , e-z+1)n, , terefore from (2),, , we, , rom)rn)=, =, , +n, , -, , 1, , dz, , r(m +n), , =, , .(2), (See $ 4-3), , (1+yn+n, , get, , ( m +n) " *idt, (1+1m tn, , (m, , +, , n), , J, , (1, , +yn, , + t, , =r(m + n) B (m,n), , B(m,n), , =, , r(m) r(n), , T(m +n), , Proved.

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05, , nd Gamma Fiunctions, , (1-x"*ldr =B(n1,n), , Sol., , =, , B, , (1,m), , -1(1 -x-1dr, , r ()T(n)-(m-1) !(n-1)!, m +n-1)!, , r(n +n), , (m-1)! (n-1)!, , n, , +m-1) (n + m-2)..n (n - 1) (n -2).. 2.1, , (n, , +m, , (m-1)!-1)!, , -1) (n + m-2)... (n + 1)n(n, , (m-1) (m- 2)... 3.2.1, ( +m-1) (n +m -2)... (n+1)n, , -, , 1)!, , Example 5. Obtain the value of the integral, T, , Jxsinfx cosx d, Sol. Let I, , x sin°x cos x d, , ), , JT x) sin' (t -x) cos (7- ) dr, Or, , I=, , Adding (i), , J ( -x) sin°x cos x dt, , and, , (i), , we, , by property 4 of definite integral), , ..(ii), , get, , 2=sinf xcos' xdr, 27, , 2t, , sinx cos*x dr, , (by property 6 of definite integral), , 2r (6), , 2.5.4.3.2.1, 256, , 512, , Example 6. Show that, Dolution. Putting 2x =t, we, , fede=, get, , 8

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ana Gammna Functions, , 107, , r/2, , sinp, , sin cos p do, 1 T(1/2) r(5/2), r(3), , V(3/2) V, , 37T, , 2-2.1, , Example 10., , Evaluate, , Solution, , J, , 16V2, , VI-*5 de, , I-F de =a-4nd, putting x*, =t, , x =t4, , or, , so, , that dx, , (1-/2.3/4 dt, , =t3/4 dt, , 3/4 (1- )1/2 dt, V4-1, , (1-3/2-1 dt, , =*, , =1(T(1/4) r(3/2))4, , -, , ., , r, , Example 11. Show that, , r, , t/2, , Sin y, , ,, , V(Sin e) d6 = z., , de, T/2, V(Sin 6 J, (sin 6)- 1/2 cos° e de, , Solution. J, , T1/4)(3/42, T(/2), 2r(3/4, d, , n/2, , V(sin ), , d6, , =, , (sin 0)1/2, , r (3/4) T(1/2), , cos de, e, , 2r(5/4), , .(ii), , Multiplying (i) and (ii), we get, , a/2, , a/2, , V(Sin 6) J, X, , Vsin @) de, , =, , 4)r(v2)r (3/4) T(12), , 2r(3/4), , 2T(5/4