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11: 28.14 Fourier Series
§28.14 Fourier Series
28.14.2 ce ν ( z , q ) = m = c 2 m ν ( q ) cos ( ν + 2 m ) z ,
28.14.3 se ν ( z , q ) = m = c 2 m ν ( q ) sin ( ν + 2 m ) z ,
Ambiguities in sign are resolved by (28.14.9) when q = 0 , and by continuity for other values of q . … For changes of sign of ν , q , and m , …
12: 36.2 Catastrophes and Canonical Integrals
36.2.19 Ψ 2 ( 0 , y ) = π 2 | y | 2 exp ( i y 2 8 ) ( exp ( i π 8 ) J 1 / 4 ( y 2 8 ) sign ( y ) exp ( i π 8 ) J 1 / 4 ( y 2 8 ) ) .
13: 1.3 Determinants, Linear Operators, and Spectral Expansions
An alternant is a determinant function of n variables which changes sign when two of the variables are interchanged. …
14: 28.2 Definitions and Basic Properties
Change of Sign of q
§28.2(vi) Eigenfunctions
the ambiguity of sign being resolved by (28.2.29) when q = 0 and by continuity for the other values of q . … For change of sign of q (compare (28.2.4)) …
15: 33.14 Definitions and Basic Properties
The choice of sign in the last line of (33.14.6) is immaterial: the same function f ( ϵ , ; r ) is obtained. …
16: 10.23 Sums
If 𝒞 = J and the upper signs are taken, then the restriction on λ is unnecessary. …
17: Bibliography K
  • W. Kahan (1987) Branch Cuts for Complex Elementary Functions or Much Ado About Nothing’s Sign Bit. In The State of the Art in Numerical Analysis (Birmingham, 1986), A. Iserles and M. J. D. Powell (Eds.), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 9, pp. 165–211.
  • 18: 7.5 Interrelations
    §7.5 Interrelations
    and either all upper signs or all lower signs are taken throughout. …
    7.5.10 g ( z ) ± i f ( z ) = 1 2 ( 1 ± i ) e ζ 2 erfc ζ .
    19: 4.21 Identities
    4.21.1 sin u ± cos u = 2 sin ( u ± 1 4 π ) = ± 2 cos ( u 1 4 π ) .
    20: 19.21 Connection Formulas
    The complete cases of R F and R G have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). Upper signs apply if 0 < ph z < π , and lower signs if π < ph z < 0 : …
    19.21.12 ( p x ) R J ( x , y , z , p ) + ( q x ) R J ( x , y , z , q ) = 3 R F ( x , y , z ) 3 R C ( ξ , η ) ,