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1: 23.15 Definitions
k = θ 2 2 ( 0 , q ) θ 3 2 ( 0 , q ) ,
The set of all bilinear transformations of this form is denoted by SL ( 2 , ) (Serre (1973, p. 77)). 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). …
2: 10.29 Recurrence Relations and Derivatives
With 𝒵 ν ( z ) defined as in §10.25(ii),
𝒵 ν - 1 ( z ) - 𝒵 ν + 1 ( z ) = ( 2 ν / z ) 𝒵 ν ( z ) ,
𝒵 ν - 1 ( z ) + 𝒵 ν + 1 ( z ) = 2 𝒵 ν ( z ) .
For k = 0 , 1 , 2 , , …
10.29.5 𝒵 ν ( k ) ( z ) = 1 2 k ( 𝒵 ν - k ( z ) + ( k 1 ) 𝒵 ν - k + 2 ( z ) + ( k 2 ) 𝒵 ν - k + 4 ( z ) + + 𝒵 ν + k ( z ) ) .
3: 15.17 Mathematical Applications
First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL ( 2 , ) , and spherical functions on certain nonsymmetric Gelfand pairs. …
4: 23.18 Modular Transformations
and λ ( τ ) is a cusp form of level zero for the corresponding subgroup of SL ( 2 , ) . … J ( τ ) is a modular form of level zero for SL ( 2 , ) . …
23.18.5 η ( 𝒜 τ ) = ε ( 𝒜 ) ( - i ( c τ + d ) ) 1 / 2 η ( τ ) ,
23.18.7 s ( d , c ) = r = 1 c - 1 r c ( d r c - d r c - 1 2 ) , c > 0 .
Note that η ( τ ) is of level 1 2 . …
5: 10.36 Other Differential Equations
The quantity λ 2 in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by - λ 2 if at the same time the symbol 𝒞 in the given solutions is replaced by 𝒵 . …
10.36.1 z 2 ( z 2 + ν 2 ) w ′′ + z ( z 2 + 3 ν 2 ) w - ( ( z 2 + ν 2 ) 2 + z 2 - ν 2 ) w = 0 , w = 𝒵 ν ( z ) ,
10.36.2 z 2 w ′′ + z ( 1 ± 2 z ) w + ( ± z - ν 2 ) w = 0 , w = e z 𝒵 ν ( z ) .
6: 33.22 Particle Scattering and Atomic and Molecular Spectra
With e denoting here the elementary charge, the Coulomb potential between two point particles with charges Z 1 e , Z 2 e and masses m 1 , m 2 separated by a distance s is V ( s ) = Z 1 Z 2 e 2 / ( 4 π ε 0 s ) = Z 1 Z 2 α c / s , where Z j are atomic numbers, ε 0 is the electric constant, α is the fine structure constant, and is the reduced Planck’s constant. … In these applications, the Z -scaled variables r and ϵ are more convenient.
Z Scaling
The Z -scaled variables r and ϵ of §33.14 are given by … For Z 1 Z 2 = - 1 and m = m e , the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, a 0 = / ( m e c α ) , and to a multiple of the Rydberg constant, …
7: 25.3 Graphics
See accompanying text
Figure 25.3.2: Riemann zeta function ζ ( x ) and its derivative ζ ( x ) , - 12 x - 2 . Magnify
See accompanying text
Figure 25.3.4: Z ( t ) , 0 t 50 . Z ( t ) and ζ ( 1 2 + i t ) have the same zeros. … Magnify
See accompanying text
Figure 25.3.5: Z ( t ) , 1000 t 1050 . Magnify
See accompanying text
Figure 25.3.6: Z ( t ) , 10000 t 10050 . Magnify
8: 35.2 Laplace Transform
For any complex symmetric matrix Z , … Then (35.2.1) converges absolutely on the region ( Z ) > X 0 , and g ( Z ) is a complex analytic function of all elements z j , k of Z . … where the integral is taken over all Z = U + i V such that U > X 0 and V ranges over 𝒮 . … If g j is the Laplace transform of f j , j = 1 , 2 , then g 1 g 2 is the Laplace transform of the convolution f 1 * f 2 , where …
9: 25.10 Zeros
Calculations relating to the zeros on the critical line make use of the real-valued function …is chosen to make Z ( t ) real, and ph Γ ( 1 4 + 1 2 i t ) assumes its principal value. Because | Z ( t ) | = | ζ ( 1 2 + i t ) | , Z ( t ) vanishes at the zeros of ζ ( 1 2 + i t ) , which can be separated by observing sign changes of Z ( t ) . Because Z ( t ) changes sign infinitely often, ζ ( 1 2 + i t ) has infinitely many zeros with t real. … Sign changes of Z ( t ) are determined by multiplying (25.9.3) by exp ( i ϑ ( t ) ) to obtain the Riemann–Siegel formula: …
10: 35.4 Partitions and Zonal Polynomials
For any partition κ , the zonal polynomial Z κ : 𝒮 is defined by the properties
35.4.2 Z κ ( I ) = | κ | !  2 2 | κ | [ m / 2 ] κ 1 j < l ( κ ) ( 2 k j - 2 k l - j + l ) j = 1 ( κ ) ( 2 k j + ( κ ) - j ) !
Therefore Z κ ( T ) is a symmetric polynomial in the eigenvalues of T . … For k = 0 , 1 , 2 , , … For T Ω and ( a ) , ( b ) > 1 2 ( m - 1 ) , …