Page 701 - Handbook Of Integral Equations

Differentiation formulas

d 1 d 1 d 1 d 1
arcsin x = √ , arccos x = – √ , arctan x = , arccot x = – .
dx 1–x 2 dx 1–x 2 dx 1+x 2 dx 1+x 2
Expansion into power series
1 x 3 1 ⋅ 3 x 5 1 ⋅ 3 ⋅ 5 x 7
arcsin x = x + + + + ··· (|x| < 1),
2 3 2 ⋅ 4 5 2 ⋅ 4 ⋅ 6 7
x 3 x 5 x 7
arctan x = x – + – + ··· (|x|≤ 1),
3 5 7
π 1 1 1
arctan x = – + – + ··· (|x| > 1).
2 x 3x 3 5x 5
The expansions for arccos x and arccot x can be obtained with the aid of the formulas arccos x =
π π
2 – arcsin x and arccot x = 2 – arctan x.

1.4. Inverse Hyperbolic Functions

Relationship with logarithmic function

√ 1+ x
1
Arsinh x =ln x + x +1 , Artanh x = ln ,
2
2 1 – x
√
2 1 1+ x
Arcosh x =ln x + x – 1 , Arcoth x = ln ;
2 x – 1
Arsinh(–x)= –Arsinh x, Artanh(–x)= –Artanh x,
Arcosh(–x) = Arcosh x, Arcoth(–x)= –Arcoth x.
Relations between inverse hyperbolic functions
√ x
2
Arsinh x = Arcosh x + 1 = Artanh √ ,
2
x +1
√
2
√ x – 1
2
Arcosh x = Arsinh x – 1 = Artanh ,
x
x 1 1
Artanh x = Arsinh √ = Arcosh √ = Arcoth .
1 – x 2 1 – x 2 x
Addition and subtraction of inverse hyperbolic functions
√

2
Arsinh x ± Arsinh y = Arsinh x 1+ y ± y 1+ x 2 ,

2 2
Arcosh x ± Arcosh y = Arcosh xy ± (x – 1)(y – 1) ,

2 2
Arsinh x ± Arcosh y = Arsinh xy ± (x + 1)(y – 1) ,
x ± y xy ± 1
Artanh x ± Artanh y = Artanh , Artanh x ± Arcoth y = Artanh .
1 ± xy y ± x
Differentiation formulas
d 1 d 1
Arsinh x = √ , Arcosh x = √ ,
dx x +1 dx x – 1
2
2
d 1 2 d 1 2
Artanh x = (x < 1), Arcoth x = (x > 1).
dx 1 – x 2 dx 1 – x 2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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