open disks around infinity
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21: 18.39 Applications in the Physical Sciences
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►where the orthogonality measure is now ,
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►Orthogonality, with measure for , for fixed
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►normalized with measure , .
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►is tridiagonalized in the complete non-orthogonal (with measure , ) basis of Laguerre functions:
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►which maps onto .
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22: Mathematical Introduction
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complex plane (excluding infinity). | |
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is finite, or converges. | |
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open interval in , or open straight-line segment joining and in . | |
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or | half-closed intervals. |
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or | matrix with th element or . |
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23: 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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►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group.
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24: 31.6 Path-Multiplicative Solutions
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►with , but with another set of .
This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the -plane that encircles and once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor .
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25: 18.16 Zeros
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►Let , , denote the zeros of as function of with
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►Then as , with () and () fixed,
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►As , with and fixed,
…when .
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►All zeros of lie in the open interval .
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26: 26.9 Integer Partitions: Restricted Number and Part Size
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►It is also equal to the number of lattice paths from to that have exactly vertices , , , above and to the left of the lattice path.
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26.9.5
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26.9.6
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26.9.7
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►As with fixed,
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27: 8.13 Zeros
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►For asymptotic approximations for and as see Tricomi (1950b), with corrections by Kölbig (1972b).
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28: 22.18 Mathematical Applications
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►With the mapping gives a conformal map of the closed rectangle onto the half-plane , with mapping to respectively.
The half-open rectangle maps onto cut along the intervals and .
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►For any two points and on this curve, their sum
, always a third point on the curve, is defined by the Jacobi–Abel addition law
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29: 17.12 Bailey Pairs
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17.12.1
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►A sequence of pairs of rational functions of several variables , , is called a Bailey pair provided that for each
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►If is a Bailey pair, then
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17.12.4
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►If is a Bailey pair, then so is , where
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30: 5.3 Graphics
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