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11: 11.9 Lommel Functions
§11.9 Lommel Functions
… ► … ►§11.9(ii) Expansions in Series of Bessel Functions
… ►For uniform asymptotic expansions, for large and fixed , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … ►12: 5.2 Definitions
…
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§5.2(i) Gamma and Psi Functions
►Euler’s Integral
… ►It is a meromorphic function with no zeros, and with simple poles of residue at . … ►
5.2.2
.
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5.2.3
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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: 4.2 Definitions
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►Most texts extend the definition of the principal value to include the branch cut
…
►In contrast to (4.2.5) the closed definition is symmetric.
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§4.2(ii) Logarithms to a General Base
… ►§4.2(iii) The Exponential Function
… ►§4.2(iv) Powers
…15: 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.
…
►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
16: 12.14 The Function
§12.14 The Function
… ►§12.14(vi) Integral Representations
… ►§12.14(vii) Relations to Other Functions
►Bessel Functions
… ►Confluent Hypergeometric Functions
…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.
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►These notations are due to Miller (1952, 1955).
An older notation, due to Whittaker (1902), for is .
The notations are related by .
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18: 11.10 Anger–Weber Functions
§11.10 Anger–Weber Functions
… ►§11.10(vi) Relations to Other Functions
… ► … ►where the prime on the second summation symbols means that the first term is to be halved. ►§11.10(ix) Recurrence Relations and Derivatives
…19: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
… ►However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of , , and , and also to complex values. … ►§8.17(ii) Hypergeometric Representations
… ►§8.17(iv) Recurrence Relations
… ►§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties
…20: 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. |