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1: 28.12 Definitions and Basic Properties
28.12.2 λ ν ( q ) = λ ν ( q ) = λ ν ( q ) .
28.12.10 me ν ( z , q ) ¯ = me ν ¯ ( z ¯ , q ¯ ) .
28.12.15 se ν ( z , q ) = se ν ( z , q ) = se ν ( z , q ) .
2: 28.5 Second Solutions fe n , ge n
S 2 m + 2 ( q ) = S 2 m + 2 ( q ) .
3: 28.2 Definitions and Basic Properties
Change of Sign of q
28.2.37 se 2 n + 2 ( z , q ) = ( 1 ) n se 2 n + 2 ( 1 2 π z , q ) .
4: 28.4 Fourier Series
§28.4(v) Change of Sign of q
5: 28.31 Equations of Whittaker–Hill and Ince
ℎ𝑠 2 n + 2 2 m + 2 ( z , ξ ) = ( 1 ) m ℎ𝑠 2 n + 2 2 m + 2 ( 1 2 π z , ξ ) .
6: 31.8 Solutions via Quadratures
The curve Γ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for m j . …
7: 10.68 Modulus and Phase Functions
§10.68(ii) Basic Properties
ϕ ν ( x ) = ϕ ν ( x ) + ν π .
§10.68(iv) Further Properties
Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …
8: 10.61 Definitions and Basic Properties
§10.61 Definitions and Basic Properties
Most properties of ber ν x , bei ν x , ker ν x , and kei ν x follow straightforwardly from the above definitions and results given in preceding sections of this chapter. …
§10.61(iii) Reflection Formulas for Arguments
§10.61(iv) Reflection Formulas for Orders
9: 11.9 Lommel Functions
Reflection Formulas
For descriptive properties of s μ , ν ( x ) see Steinig (1972). …
10: 4.37 Inverse Hyperbolic Functions
4.37.6 Arccoth z = Arctanh ( 1 / z ) .
§4.37(iii) Reflection Formulas
§4.37(v) Fundamental Property