orthogonal functions with respect to weighted summation
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11: 16.13 Appell Functions
§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
,
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16.13.4
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12: 5.12 Beta Function
§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 . …13: 31.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►The main functions treated in this chapter are , , , and the polynomial .
…Sometimes the parameters are suppressed.
, | real variables. |
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14: 10.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►For the spherical Bessel functions and modified spherical Bessel functions the order is a nonnegative integer.
For the other functions when the order is replaced by , it can be any integer.
For the Kelvin functions the order is always assumed to be real.
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►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
15: 4.2 Definitions
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►Most texts extend the definition of the principal value to include the branch cut
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§4.2(ii) Logarithms to a General Base
… ►§4.2(iii) The Exponential Function
… ►§4.2(iv) Powers
… ►Again, without the closed definition the and signs would have to be replaced by and , respectively.16: 25.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►The main function treated in this chapter is the Riemann zeta function
.
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►The main related functions are the Hurwitz zeta function
, the dilogarithm , the polylogarithm (also known as Jonquière’s function
), Lerch’s transcendent , and the Dirichlet -functions
.
nonnegative integers. | |
… | |
primes | on function symbols: derivatives with respect to argument. |
17: 12.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values.
►The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: , , , and .
These notations are due to Miller (1952, 1955).
An older notation, due to Whittaker (1902), for is .
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