mean value property for harmonic functions
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1: 19.8 Quadratic Transformations
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§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)
… ►As , and converge to a common limit called the AGM (Arithmetic-Geometric Mean) of and . …showing that the convergence of to 0 and of and to is quadratic in each case. … ►Again, and converge quadratically to and 0, respectively, and converges to 0 faster than quadratically. If , then the Cauchy principal value is …2: 14.30 Spherical and Spheroidal Harmonics
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§14.30(ii) Basic Properties
►Most mathematical properties of can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. … ►Special Values
… ►3: 9.12 Scorer Functions
§9.12 Scorer Functions
… ►§9.12(iii) Initial Values
… ►where the last integral is a Cauchy principal value (§1.4(v)). … ► … ►For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c). …4: 5.2 Definitions
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§5.2(i) Gamma and Psi Functions
►Euler’s Integral
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5.2.1
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►It is a meromorphic function with no zeros, and with simple poles of residue at .
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5.2.3
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5: 5.15 Polygamma Functions
§5.15 Polygamma Functions
►The functions , , are called the polygamma functions. In particular, is the trigamma function; , , are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … ►For see §24.2(i). …6: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
… ► ►For all values of … ►§15.2(ii) Analytic Properties
… ►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. …7: 11.9 Lommel Functions
§11.9 Lommel Functions
… ►Another solution of (11.9.1) that is defined for all values of and is , where … … ►For descriptive properties of see Steinig (1972). … ►8: 23.15 Definitions
§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, … … ►Dedekind’s Eta Function (or Dedekind Modular Function)
… ►9: 5.12 Beta Function
§5.12 Beta Function
►In this section all fractional powers have their principal values, except where noted otherwise. … ►In (5.12.8) the fractional powers have their principal values when and , and are continued via continuity. … ►In (5.12.11) and (5.12.12) the fractional powers are continuous on the integration paths and take their principal values at the beginning. … ►Pochhammer’s Integral
…10: 14.20 Conical (or Mehler) Functions
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►Another real-valued solution of (14.20.1) was introduced in Dunster (1991).
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►Lastly, for the range , is a real-valued solution of (14.20.1); in terms of (which are complex-valued in general):
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►where the inverse trigonometric functions take their principal values.
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►with the inverse tangent taking its principal value.
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