Begin2.DVI

∫ J 1 (x) x dx=−J^1 (x) +

∫ J 0 (x)dx

∫ xνJν− 1 (x)dx=xνJν(x) +C

∫ x−νJν+1(x)dx=x−νJν(x) +C

∫ J 1 (x) xn dx=

− 1 n

J 1 (x) xn−^1 +

1 n

∫ J 0 (x) xn−^1 dx

∫ xJ 0 (x)dx=xJ 1 (x) +C

∫ x^2 J 0 (x)dx=x^2 J 1 (x) +xJ 0 (x)−

∫ J 0 (x)dx

∫ xnJ 0 (x)dx=xnJ 1 (x) + (n−1)xn−^1 J 0 (x)−(n−1)^2

∫ xn−^2 J 0 (x)dx

∫ J 0 (x) xn dx=

J 1 (x) (n−1)^2 xn−^2 −

J 0 (x) (n−1)xn−^1 −

1 (n−1)^2

∫ J 0 (x) xn−^2 dx

∫ Jn+1(x)dx=

∫ Jn− 1 (x)dx− 2 Jn(x)

∫

xJn(αx)Jn(βx)dx=β (^2) −xα 2 [αJn′(αx)Jn(βx)−βJn′(βx)Jn(αx)] +C

IfIm,n=

∫ xmJn(x)dx, m≥−n, then

Im,n=−xmJn− 1 (x) + (m+n−1)Im− 1 ,n− 1

IfIn, 0 =

∫ xnJ 0 (x)dx, thenIn, 0 =xnJ 1 (x) + (n−1)xn−^1 J 0 (x)−(n−1)^2 In− 2 , 0 Note that

I 1 , 0 =

∫ xJ 0 (x)dx=xJ 1 (x) +CandI 0 , 1 =

∫ J 1 (x)dx=−J 0 (x) +C Note also that the integral

I 0 , 0 =

∫ J 0 (x)dxcannot be given in closed form.

Appendix C