Page 699 - Handbook Of Integral Equations

Powers of hyperbolic functions
2
2
1
1
cosh x = 1 cosh 2x + , sinh x = 1 cosh 2x – ,
2 2 2 2
3
3
cosh x = 1 cosh 3x + 3 cosh x, sinh x = 1 sinh 3x – 3 sinh x,
4 4 4 4
4
3
3
4
cosh x = 1 cosh 4x + 1 cosh 2x + , sinh x = 1 cosh 4x – 1 cosh 2x + ,
8 2 8 8 2 8
5
5
cosh x = 1 cosh 5x + 5 cosh 3x + 5 cosh x, sinh x = 1 sinh 5x – 5 sinh 3x + 5 sinh x,
16 16 8 16 16 8
n–1
1 k 1 n
2n
cosh x = C 2n cosh[2(n – k)x]+ C ,
2n
2 2n–1 2 2n
k=0
n
1 k
2n+1
cosh x = C 2n+1 cosh[(2n – 2k +1)x],
2 2n
k=0
n–1 n
1 k k (–1) n
2n
sinh x = (–1) C 2n cosh[2(n – k)x]+ C ,
2n
2 2n–1 2 2n
k=0
n
1 k k
2n+1
sinh x = (–1) C 2n+1 sinh[(2n – 2k +1)x].
2 2n
k=0
k
Here C m are binomial coefficients.
Hyperbolic functions of multiple arguments
2
cosh 2x = 2 cosh x – 1, sinh 2x = 2 sinh x cosh x,
3
3
cosh 3x = –3 cosh x + 4 cosh x, sinh 3x = 3 sinh x + 4 sinh x,
3
4
2
cosh 4x =1 – 8 cosh x + 8 cosh x, sinh 4x = 4 cosh x(sinh x + 2 sinh x),
3
5
3
5
cosh 5x = 5 cosh x – 20 cosh x + 16 cosh x, sinh 5x = 5 sinh x + 20 sinh x + 16 sinh x.
[n/2] k+1
n (–1) k–2 n–2k–2 n–2k–2
n
n–1
cosh(nx)=2 cosh x + C n–k–2 2 (cosh x) ,
2 k +1
k=0
[(n–1)/2]
n–k–1 k n–2k–1
sinh(nx) = sinh x 2 C n–k–1 (cosh x) .
k=0
k
Here C m are binomial coefficients and [A] stands for the integer part of a number A.
Relationship with trigonometric functions
2
sinh(ix)= i sin x, cosh(ix) = cos x, tanh(ix)= i tan x, coth(ix)= –i cot x, i = –1.
Differentiation formulas
d sinh x d cosh x d tanh x 1 d coth x 1
= cosh x, = sinh x, = , = – .
2
2
dx dx dx cosh x dx sinh x
Expansion into power series
x 2 x 4 x 6
cosh x =1 + + + + ··· (|x| < ∞),
2! 4! 6!
x 3 x 5 x 7
sinh x = x + + + + ··· (|x| < ∞),
3! 5! 7!
x 3 2x 5 17x 7
tanh x = x – + – + ··· (|x| < π/2),
3 15 315
1 x x 3 2x 5
coth x = + – + – ··· (|x| < π).
x 3 45 945
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 683