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real and imaginary parts

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11: 4.8 Identities
where the integer k is chosen so that ( i z ln a ) + 2 k π [ π , π ] . …
12: 25.4 Reflection Formulas
25.4.5 ( 1 ) k ζ ( k ) ( 1 s ) = 2 ( 2 π ) s m = 0 k r = 0 m ( k m ) ( m r ) ( ( c k m ) cos ( 1 2 π s ) + ( c k m ) sin ( 1 2 π s ) ) Γ ( r ) ( s ) ζ ( m r ) ( s ) ,
13: 10.14 Inequalities; Monotonicity
| J ν ( x ) | 1 , ν 0 , x ,
| J ν ( x ) | 2 1 2 , ν 1 , x .
10.14.4 | J ν ( z ) | | 1 2 z | ν e | z | Γ ( ν + 1 ) , ν 1 2 .
10.14.7 1 J ν ( ν x ) x ν J ν ( ν ) e ν ( 1 x ) , ν 0 , 0 < x 1 ;
14: 23.15 Definitions
In §§23.1523.19, k and k ( ) denote the Jacobi modulus and complementary modulus, respectively, and q = e i π τ ( τ > 0 ) denotes the nome; compare §§20.1 and 22.1. … 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 , ) ,
23.15.5 f ( 𝒜 τ ) = c 𝒜 ( c τ + d ) f ( τ ) , τ > 0 ,
In (23.15.9) the branch of the cube root is chosen to agree with the second equality; in particular, when τ lies on the positive imaginary axis the cube root is real and positive. …
15: 20.13 Physical Applications
with κ = i π / 4 . For τ = i t , with α , t , z real, (20.13.1) takes the form of a real-time t diffusion equation …Let z , α , t . …
20.13.4 π 4 α t n = e ( n π + z ) 2 / ( 4 α t ) = θ 3 ( z | i 4 α t / π ) ,
In the singular limit τ 0 + , the functions θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …
16: 28.17 Stability as x ±
If all solutions of (28.2.1) are bounded when x ± along the real axis, then the corresponding pair of parameters ( a , q ) is called stable. … For example, positive real values of a with q = 0 comprise stable pairs, as do values of a and q that correspond to real, but noninteger, values of ν . However, if ν 0 , then ( a , q ) always comprises an unstable pair. …Also, all nontrivial solutions of (28.2.1) are unbounded on . For real a and q ( 0 ) the stable regions are the open regions indicated in color in Figure 28.17.1. …
17: 28.9 Zeros
There are no zeros within the strip | z | < 1 2 π other than those on the real and imaginary axes. …
18: 15.17 Mathematical Applications
The quotient of two solutions of (15.10.1) maps the closed upper half-plane z 0 conformally onto a curvilinear triangle. … 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. …
19: 22.3 Graphics
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Figure 22.3.25: sn ( 5 , k ) as a function of complex k 2 , 1 ( k 2 ) 3.5 , 1 ( k 2 ) 1 . … Magnify 3D Help
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Figure 22.3.26: Density plot of | sn ( 5 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
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Figure 22.3.27: Density plot of | sn ( 10 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
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Figure 22.3.28: Density plot of | sn ( 20 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
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Figure 22.3.29: Density plot of | sn ( 30 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . … Magnify
20: 4.45 Methods of Computation
4.45.16 e z = e z ( cos ( z ) + i sin ( z ) ) .