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solutions as trigonometric and hyperbolic functions

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1: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
Arcsinh z and Arccsch z have branch points at z = ± i ; the other four functions have branch points at z = ± 1 . … Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … For the corresponding results for arccsch z , arcsech z , and arccoth z , use (4.37.7)–(4.37.9); compare §4.23(iv). …
§4.37(vi) Interrelations
2: 4.34 Derivatives and Differential Equations
4.34.12 w = ( 1 / a ) sinh ( a z + c ) ,
4.34.13 w = ( 1 / a ) cosh ( a z + c ) ,
4.34.14 w = ( 1 / a ) coth ( a z + c ) ,
3: 4.43 Cubic Equations
§4.43 Cubic Equations
4: 14.19 Toroidal (or Ring) Functions
§14.19 Toroidal (or Ring) Functions
§14.19(i) Introduction
When ν = n 1 2 , n = 0 , 1 , 2 , , μ , and x ( 1 , ) solutions of (14.2.2) are known as toroidal or ring functions. …
§14.19(iv) Sums
§14.19(v) Whipple’s Formula for Toroidal Functions
5: 28.32 Mathematical Applications
28.32.3 2 V ξ 2 + 2 V η 2 + 1 2 c 2 k 2 ( cosh ( 2 ξ ) cos ( 2 η ) ) V = 0 .
6: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
§4.23(i) General Definitions
§4.23(ii) Principal Values
§4.23(iv) Logarithmic Forms
§4.23(vii) Special Values and Interrelations
7: 28.28 Integrals, Integral Representations, and Integral Equations
28.28.16 0 sin ( 2 h cos y cosh t ) Ce 2 n ( t , h 2 ) d t = π A 0 2 n ( h 2 ) 2 ce 2 n ( 1 2 π , h 2 ) ( ce 2 n ( y , h 2 ) 2 π C 2 n ( h 2 ) fe 2 n ( y , h 2 ) ) ,
8: 28.20 Definitions and Basic Properties
28.20.1 w ′′ ( a 2 q cosh ( 2 z ) ) w = 0 ,
28.20.2 ( ζ 2 1 ) w ′′ + ζ w + ( 4 q ζ 2 2 q a ) w = 0 , ζ = cosh z .
9: 10.24 Functions of Imaginary Order
§10.24 Functions of Imaginary Order
and J ~ ν ( x ) , Y ~ ν ( x ) are linearly independent solutions of (10.24.1): … In consequence of (10.24.6), when x is large J ~ ν ( x ) and Y ~ ν ( x ) comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). Also, in consequence of (10.24.7)–(10.24.9), when x is small either J ~ ν ( x ) and tanh ( 1 2 π ν ) Y ~ ν ( x ) or J ~ ν ( x ) and Y ~ ν ( x ) comprise a numerically satisfactory pair depending whether ν 0 or ν = 0 . …
10: 14.15 Uniform Asymptotic Approximations
See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials P n ( cos θ ) as n with θ fixed. … where the inverse trigonometric functions take their principal values (§4.23(ii)). …The interval 0 x < is mapped one-to-one to the interval < y y 0 , where y = y 0 is the (positive) solution of (14.15.21) when x = 0 . … The inverse hyperbolic and trigonometric functions take their principal values (§§4.23(ii), 4.37(ii)). … (The inverse hyperbolic functions again take their principal values.) …