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hyperbolic cosecant function

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1: 4.1 Special Notation
k , m , n integers.
; the hyperbolic trigonometric (or just hyperbolic) functions sinh z , cosh z , tanh z , csch z , sech z , coth z ; the inverse hyperbolic functions arcsinh z , Arcsinh z , etc. …
2: 4.28 Definitions and Periodicity
4.28.5 csch z = 1 sinh z ,
4.28.11 csc ( i z ) = i csch z ,
3: 4.29 Graphics
§4.29(i) Real Arguments
See accompanying text
Figure 4.29.5: csch x and sech x . Magnify
§4.29(ii) Complex Arguments
The conformal mapping w = sinh z is obtainable from Figure 4.15.7 by rotating both the w -plane and the z -plane through an angle 1 2 π , compare (4.28.8). The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …
4: 4.36 Infinite Products and Partial Fractions
4.36.4 csch 2 z = n = 1 ( z n π i ) 2 ,
4.36.5 csch z = 1 z + 2 z n = 1 ( 1 ) n z 2 + n 2 π 2 .
5: 4.31 Special Values and Limits
§4.31 Special Values and Limits
Table 4.31.1: Hyperbolic functions: values at multiples of 1 2 π i .
z 0 1 2 π i π i 3 2 π i
4.31.1 lim z 0 sinh z z = 1 ,
4.31.2 lim z 0 tanh z z = 1 ,
6: 4.34 Derivatives and Differential Equations
4.34.4 d d z csch z = csch z coth z ,
4.34.6 d d z coth z = csch 2 z .
7: 4.18 Inequalities
4.18.7 | csc z | csch | y | ,
8: 4.30 Elementary Properties
§4.30 Elementary Properties
Table 4.30.1: Hyperbolic functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.
sinh θ = a cosh θ = a tanh θ = a csch θ = a sech θ = a coth θ = a
csch θ a 1 ( a 2 1 ) 1 / 2 a 1 ( 1 a 2 ) 1 / 2 a a ( 1 a 2 ) 1 / 2 ( a 2 1 ) 1 / 2
9: 4.37 Inverse Hyperbolic Functions
4.37.4 Arccsch z = Arcsinh ( 1 / z ) ,
Arcsinh z and Arccsch z have branch points at z = ± i ; the other four functions have branch points at z = ± 1 . …
4.37.7 arccsch z = arcsinh ( 1 / z ) ,
4.37.13 arccsch ( z ) = arccsch z .
For the corresponding results for arccsch z , arcsech z , and arccoth z , use (4.37.7)–(4.37.9); compare §4.23(iv). …
10: 4.40 Integrals
4.40.4 csch x d x = ln ( tanh ( 1 2 x ) ) , 0 < x < .
4.40.14 arccsch x d x = x arccsch x + arcsinh x , 0 < x < ,