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1: 23.15 Definitions
§23.15 Definitions
A modular function f ( τ ) is a function of τ that is meromorphic in the half-plane τ > 0 , and has the property that for all 𝒜 SL ( 2 , ) , or for all 𝒜 belonging to a subgroup of SL ( 2 , ) , …(Some references refer to 2 as the level). …If, in addition, f ( τ ) 0 as q 0 , then f ( τ ) is called a cusp form. …
2: 14.19 Toroidal (or Ring) Functions
§14.19 Toroidal (or Ring) Functions
§14.19(i) Introduction
This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates ( η , θ , ϕ ) , which are related to Cartesian coordinates ( x , y , z ) by …
§14.19(iv) Sums
§14.19(v) Whipple’s Formula for Toroidal Functions
3: 9.12 Scorer Functions
§9.12 Scorer Functions
- Gi ( x ) is a numerically satisfactory companion to the complementary functions Ai ( x ) and Bi ( x ) on the interval 0 x < . Hi ( x ) is a numerically satisfactory companion to Ai ( x ) and Bi ( x ) on the interval - < x 0 . … As z , and with δ denoting an arbitrary small positive constant, …
4: 9.1 Special Notation
(For other notation see Notation for the Special Functions.)
k

nonnegative integer, except in §9.9(iii).

primes

derivatives with respect to argument.

The main functions treated in this chapter are the Airy functions Ai ( z ) and Bi ( z ) , and the Scorer functions Gi ( z ) and Hi ( z ) (also known as inhomogeneous Airy functions). Other notations that have been used are as follows: Ai ( - x ) and Bi ( - x ) for Ai ( x ) and Bi ( x ) (Jeffreys (1928), later changed to Ai ( x ) and Bi ( x ) ); U ( x ) = π Bi ( x ) , V ( x ) = π Ai ( x ) (Fock (1945)); A ( x ) = 3 - 1 / 3 π Ai ( - 3 - 1 / 3 x ) (Szegő (1967, §1.81)); e 0 ( x ) = π Hi ( - x ) , e ~ 0 ( x ) = - π Gi ( - x ) (Tumarkin (1959)).
5: 20.2 Definitions and Periodic Properties
§20.2(i) Fourier Series
Corresponding expansions for θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , can be found by differentiating (20.2.1)–(20.2.4) with respect to z .
§20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …
§20.2(iv) z -Zeros
6: 15.2 Definitions and Analytical Properties
§15.2(i) Gauss Series
The branch obtained by introducing a cut from 1 to + on the real z -axis, that is, the branch in the sector | ph ( 1 - z ) | π , is the principal branch (or principal value) of F ( a , b ; c ; z ) . … Because of the analytic properties with respect to a , b , and c , it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. … which sometimes needs to be used in §15.4. … In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that F ( - m , b ; - m ; z ) is equal to the first m + 1 terms of the Maclaurin series for ( 1 - z ) - b .
7: 5.12 Beta Function
§5.12 Beta Function
Euler’s Beta Integral
5.12.2 0 π / 2 sin 2 a - 1 θ cos 2 b - 1 θ d θ = 1 2 B ( a , b ) .
Pochhammer’s Integral
where the contour starts from an arbitrary point P in the interval ( 0 , 1 ) , circles 1 and then 0 in the positive sense, circles 1 and then 0 in the negative sense, and returns to P . …
8: 5.15 Polygamma Functions
§5.15 Polygamma Functions
The functions ψ ( n ) ( z ) , n = 1 , 2 , , are called the polygamma functions. …Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … As z in | ph z | π - δ …For B 2 k see §24.2(i). …
9: 16.13 Appell Functions
§16.13 Appell Functions
The following four functions of two real or complex variables x and y cannot be expressed as a product of two F 1 2 functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1):
16.13.1 F 1 ( α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , max ( | x | , | y | ) < 1 ,
16.13.4 F 4 ( α , β ; γ , γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m + n ( γ ) m ( γ ) n m ! n ! x m y n , | x | + | y | < 1 .
10: 14.20 Conical (or Mehler) Functions
§14.20(iii) Behavior as x 1
As τ , … As μ , …
§14.20(x) Zeros and Integrals