relation to line broadening function
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§23.15 Definitions… ►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). …If, in addition, as , then is called a cusp form. … ►
§14.19 Toroidal (or Ring) Functions►
§14.19(i) Introduction… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates , which are related to Cartesian coordinates by … ►
§14.19(iv) Sums… ►
§14.19(v) Whipple’s Formula for Toroidal Functions…
§9.12 Scorer Functions… ► is a numerically satisfactory companion to the complementary functions and on the interval . is a numerically satisfactory companion to and on the interval . … ► … ►As , and with denoting an arbitrary small positive constant, …
… ►(For other notation see Notation for the Special Functions.) ►
►The main functions treated in this chapter are the Airy functions
and , and the Scorer functions
and (also known as inhomogeneous Airy functions).
►Other notations that have been used are as follows: and for and (Jeffreys (1928), later changed to
and ); , (Fock (1945)); (Szegő (1967, §1.81)); , (Tumarkin (1959)).
nonnegative integer, except in §9.9(iii).
derivatives with respect to argument.
§20.2(i) Fourier Series… ►Corresponding expansions for , , can be found by differentiating (20.2.1)–(20.2.4) with respect to . ►
§20.2(ii) Periodicity and Quasi-Periodicity… ►The theta functions are quasi-periodic on the lattice: … ►
§15.2(i) Gauss Series… ►The branch obtained by introducing a cut from to on the real -axis, that is, the branch in the sector , is the principal branch (or principal value) of . … ►Because of the analytic properties with respect to , , and , it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. … ►which sometimes needs to be used in §15.4. … ►In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that is equal to the first terms of the Maclaurin series for .
§5.12 Beta Function… ►
Euler’s Beta Integral… ►
Pochhammer’s Integral… ►where the contour starts from an arbitrary point in the interval , circles and then in the positive sense, circles and then in the negative sense, and returns to . …
§5.15 Polygamma Functions►The functions , , are called the polygamma functions. …Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … ►As in …For see §24.2(i). …
§16.13 Appell Functions►The following four functions of two real or complex variables and cannot be expressed as a product of two functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): ►
16.13.1 ,… ►
16.13.4 .… ► …