SL(2,Z)
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1: 23.15 Definitions
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►The set of all bilinear transformations of this form is denoted by SL
(Serre (1973, p. 77)).
►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
,
…(Some references refer to as the level).
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2: 10.29 Recurrence Relations and Derivatives
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►With defined as in §10.25(ii),
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►For results on modified quotients of the form see Onoe (1955) and Onoe (1956).
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►For ,
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3: 15.17 Mathematical Applications
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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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4: 23.18 Modular Transformations
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►and is a cusp form of level zero for the corresponding subgroup of SL
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is a modular form of level zero for SL
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23.18.5
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23.18.7
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►Note that is of level .
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5: 10.36 Other Differential Equations
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►The quantity in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by if at the same time the symbol in the given solutions is replaced by .
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10.36.1
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10.36.2
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6: 33.22 Particle Scattering and Atomic and Molecular Spectra
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►With denoting here the elementary charge, the Coulomb potential between two point particles with charges and masses separated by a distance is , where are atomic numbers, is the electric constant, is the fine structure constant, and is the reduced Planck’s constant.
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►In these applications, the -scaled variables and are more convenient.
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Scaling
►The -scaled variables and of §33.14 are given by … ►For and , the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, , and to a multiple of the Rydberg constant, …7: 25.3 Graphics
8: 35.2 Laplace Transform
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►For any complex symmetric matrix ,
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►Then (35.2.1) converges absolutely on the region , and is a complex analytic function of all elements of .
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35.2.2
►where the integral is taken over all such that and ranges over .
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►If is the Laplace transform of , , then is the Laplace transform of the convolution , where
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9: 25.10 Zeros
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►Calculations relating to the zeros on the critical line make use of the real-valued function
…is chosen to make real, and assumes its principal value.
Because , vanishes at the zeros of , which can be separated by observing sign changes of .
Because changes sign infinitely often, has infinitely many zeros with real.
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►Sign changes of are determined by multiplying (25.9.3) by to obtain the Riemann–Siegel formula:
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