Begin2.DVI

∫∞

0

sinhpx

sinhqxdx=

π

2 qtan(

πp

2 q), |p|< q

∫∞

0

coshax−coshbx

sinhπx dx= ln

∣∣

∣∣

∣

cos 2 b

cosa 2

∣∣

∣∣

∣, −π < b < a < π

∫∞

0

sinhpx

sinhqxcosmx dx=

π

2 q

sinπpq

cosπpq + coshπmq , q >^0 , p

(^2) < q 2
147.
∫∞
0
sinhpx
coshqxsinmx dx=
π
q
sinpπ 2 qsinhmπ 2 q
cospπq + coshmπq
148.
∫∞
0
coshpx
coshqxcosmx dx=
π
q
cospπ 2 qcoshmπ 2 q
cospπq + coshmπq
Miscellaneous Integrals
149.
∫x
0
ξλ−^1 [1−ξμ]νdξ=x
λ
λ F(−ν,
λ
μ;
λ
μ+ 1;x
μ) See hypergeometric function
150.
∫π
0
cos(nφ−xsinφ)dφ=π Jn(x)
151.
∫a
−a
(a+x)m−^1 (a−x)n−^1 dx= (2a)m+n−^1 Γ(Γ(mm)Γ(+nn))

152. Iff′(x)is continuous and

∫∞

1

f(x)−f(∞)

x dxconverges, then

∫∞

0

f(ax)−f(bx)

x dx= [f(0)−f(∞)] ln

b

a

153. Iff(x) =f(−x)so thatf(x)is an even function, then
     ∫∞

0

f

(

x−x^1

)

dx=

∫∞

0

f(x)dx

154. Elliptic integral of the first kind
     ∫θ

0

√ dθ

1 −k^2 sin^2 θ

=F(θ, k), 0 < k < 1

155. Elliptic integral of the second kind
     ∫θ

0

√

1 −k^2 sin^2 θ dθ=E(θ, k)

156. Elliptic integral of the third kind
     ∫θ

0

dθ

(1 +nsin^2 θ)

√

1 −k^2 sin^2 θ

= Π(θ, k, n)

Appendix C