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1: 31.2 Differential Equations
§31.2(iii) Trigonometric Form
2: 29.2 Differential Equations
29.2.4 ( 1 - k 2 cos 2 ϕ ) d 2 w d ϕ 2 + k 2 cos ϕ sin ϕ d w d ϕ + ( h - ν ( ν + 1 ) k 2 cos 2 ϕ ) w = 0 ,
3: 4.23 Inverse Trigonometric Functions
§4.23(iv) Logarithmic Forms
Care needs to be taken on the cuts, for example, if 0 < x < then 1 / ( x + i 0 ) = ( 1 / x ) - i 0 . …
4: 18.11 Relations to Other Functions
Hermite
5: 6.5 Further Interrelations
6.5.3 1 2 ( Ei ( x ) + E 1 ( x ) ) = Shi ( x ) = - i Si ( i x ) ,
6.5.4 1 2 ( Ei ( x ) - E 1 ( x ) ) = Chi ( x ) = Ci ( i x ) - 1 2 π i .
6: 6.7 Integral Representations
6.7.7 0 1 e - a t sin ( b t ) t d t = Ein ( a + i b ) , a , b ,
6.7.8 0 1 e - a t ( 1 - cos ( b t ) ) t d t = Ein ( a + i b ) - Ein ( a ) , a , b .
6.7.10 Ein ( z ) - Cin ( z ) = 0 π / 2 e - z cos t sin ( z sin t ) d t ,
6.7.11 0 1 ( 1 - e - a t ) cos ( b t ) t d t = Ein ( a + i b ) - Cin ( b ) , a , b .
7: 6.2 Definitions and Interrelations
6.2.10 si ( z ) = - z sin t t d t = Si ( z ) - 1 2 π .
6.2.13 Ci ( z ) = - Cin ( z ) + ln z + γ .
8: 28.2 Definitions and Basic Properties
With ζ = sin 2 z we obtain the algebraic form of Mathieu’s equation …With ζ = cos z we obtain another algebraic form: …
28.2.16 cos ( π ν ) = w I ( π ; a , q ) = w I ( π ; a , - q ) .
9: 22.5 Special Values
§22.5(ii) Limiting Values of k
In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. …
Table 22.5.3: Limiting forms of Jacobian elliptic functions as k 0 .
sn ( z , k ) sin z cd ( z , k ) cos z dc ( z , k ) sec z ns ( z , k ) csc z
Table 22.5.4: Limiting forms of Jacobian elliptic functions as k 1 .
sn ( z , k ) tanh z cd ( z , k ) 1 dc ( z , k ) 1 ns ( z , k ) coth z
10: 4.45 Methods of Computation
The inverses arcsinh , arccosh , and arctanh can be computed from the logarithmic forms given in §4.37(iv), with real arguments. … The trigonometric functions may be computed from the definitions (4.14.1)–(4.14.7), and their inverses from the logarithmic forms in §4.23(iv), followed by (4.23.7)–(4.23.9). …