Question

b a) Calculate the integral including Bessel functions SxJ, (ax)dx b)Calculate the integral including Legendre functions P,(x) =dx V1 - x

Answer 1

liven, 0 ana we get bg tung Recurrence Solna oral -b x5 To (ax) are Put e so that ada= ett. Then ab x5 Jolax) dx = t 15 Jo(t) dt q 0 be 25 Jolax) dx = 1 - ab es Jolt) It a6 o Relation of Bessel's We have. b C a M 0 O - us d da { xh Jake)? = xn In 1(x)

05 I an Indx an Inx (2 for Now from a ab rab 45 Jolt) dt Tab 200 #4 [+ Jolt.) olt [ 24 (+ 3, 4) = (443434 de 2 I to Integrating by parts and using (2) for ASI, (t) 46 44 ], (A) dt ws n = 1. ab of 4 +(82.), 4 ) 34.24 Sest le tao Is J, (t) - 4 12 t J . It) off?), 4) de qi.using (2) for n=2} =ts J, A 4 44 J₂ (t) + 4x2 ab. so + 3 J. (H) dt. 2 ab. to ab cotte 464 J2 Lt) + 8ft 3 Beta)

at 0 ts Jold) dt = [1965], (46) -4 (ab) 4 Jo (ab) +8 (ab)3-J3(ab) 0 nel liave. o Putting above value in a eb x5 Jolanda = 1 9565 I, (an) -4 a 464 96. J₂ (ab) + 8a363 Izlab 25 J. Cax) dx = 565 obs I, (ab) – 4 64. Jo lab] a. + 8 16 13 J3(ab) C a? a result. (6) + Which is the required Given integral Prine de 17-* Rodrigue's formula for legendre f(x) Pn(x) du f(x) Du (x2-1)ibe By using han have. Thee 1 ann! |

Where Dne In dan. We have f(x) = 1 1 Vi-se. Pube) dx | - X anni a on (x²-1) n. dx. rise -1 On inte integrating i.e. by parts we have u fvda - ( dessvdx? de u v dx 1 Dn-1 (x2-1) M 2 annillrox -1 + 2 (1-2) t an F こ 1 2 Dn-1 (²-1 (VT-x)? O (-1) +1 Dn-1 (x²-1)da ann! 2 (1-x)3/2 n's the first termên) (), vanishes on using teibnitz theorema (-1)1. Du-1 (x²-1) n dre (1-x)312 On integriting by parts again (-1) 0-[ +3 Dn-2x² 15 de antin! 2 : the first term vanishes as is previous? ght n! 2

Date Page Hz 1 DM-2(x²-1) dxco ant2 n! (1-2) On Continuing the linning the similar. n-2 times more hll have

(-in ท (x²-1)h ď Idre ann! 1 dan. Irix Ansa Idence, ( Pn(a) Pn(a) de t -un 6e2-14 dns i 20 ann! dan 1-x) de which is the required result