real and imaginary parts
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11—20 of 180 matching pages
11: 4.8 Identities
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►where the integer is chosen so that .
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12: 25.4 Reflection Formulas
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25.4.5
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13: 10.14 Inequalities; Monotonicity
14: 23.15 Definitions
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►In §§23.15–23.19, and
denote the Jacobi modulus and complementary modulus, respectively, and () denotes the nome; compare §§20.1 and 22.1.
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►A modular function
is a function of that is meromorphic in the half-plane , and has the property that for all , or for all belonging to a subgroup of SL,
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23.15.5
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23.15.9
►In (23.15.9) the branch of the cube root is chosen to agree with the second equality; in particular, when lies on the positive imaginary axis the cube root is real and positive.
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15: 20.13 Physical Applications
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►with .
►For , with
real, (20.13.1) takes the form of a real-time diffusion equation
…Let .
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20.13.4
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►In the singular limit , the functions , , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195).
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16: 28.17 Stability as
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►If all solutions of (28.2.1) are bounded when along the real axis, then the corresponding pair of parameters is called stable.
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►For example, positive real values of with comprise stable pairs, as do values of and that correspond to real, but noninteger, values of .
►However, if , then always comprises an unstable pair.
…Also, all nontrivial solutions of (28.2.1) are unbounded on .
►For real
and
the stable regions are the open regions indicated in color in Figure 28.17.1.
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17: 28.9 Zeros
18: 15.17 Mathematical Applications
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►The quotient of two solutions of (15.10.1) maps the closed upper half-plane conformally onto a curvilinear triangle.
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►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL, and spherical functions on certain nonsymmetric Gelfand pairs.
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19: 22.3 Graphics
20: 4.45 Methods of Computation
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4.45.16
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