About the Project

relation to line broadening function

AdvancedHelp

(0.005 seconds)

11—20 of 1012 matching pages

11: 11.9 Lommel Functions
§11.9 Lommel Functions
§11.9(ii) Expansions in Series of Bessel Functions
For uniform asymptotic expansions, for large ν and fixed μ = 1 , 0 , 1 , 2 , , 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
§5.2(i) Gamma and Psi Functions
Euler’s Integral
It is a meromorphic function with no zeros, and with simple poles of residue ( 1 ) n / n ! at z = n . …
5.2.2 ψ ( z ) = Γ ( z ) / Γ ( z ) , z 0 , 1 , 2 , .
5.2.3 γ = lim n ( 1 + 1 2 + 1 3 + + 1 n ln n ) = 0.57721 56649 01532 86060 .
13: 31.1 Special Notation
(For other notation see Notation for the Special Functions.)
x , y real variables.
The main functions treated in this chapter are H ( a , q ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) 𝐻𝑓 m ( a , q m ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) 𝐻𝑓 m ν ( a , q m ; α , β , γ , δ ; z ) , and the polynomial 𝐻𝑝 n , m ( a , q n , m ; n , β , γ , δ ; z ) . …Sometimes the parameters are suppressed.
14: 4.2 Definitions
Most texts extend the definition of the principal value to include the branch cutIn contrast to (4.2.5) the closed definition is symmetric. …
§4.2(ii) Logarithms to a General Base a
§4.2(iii) The Exponential Function
§4.2(iv) Powers
15: 10.1 Special Notation
(For other notation see Notation for the Special Functions.) … For the spherical Bessel functions and modified spherical Bessel functions the order n is a nonnegative integer. For the other functions when the order ν is replaced by n , it can be any integer. For the Kelvin functions the order ν 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 W ( a , x )
§12.14 The Function W ( a , x )
§12.14(vi) Integral Representations
§12.14(vii) Relations to Other Functions
Bessel Functions
Confluent Hypergeometric Functions
17: 12.1 Special Notation
(For other notation see Notation for the Special Functions.) … Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. … These notations are due to Miller (1952, 1955). An older notation, due to Whittaker (1902), for U ( a , z ) is D ν ( z ) . The notations are related by U ( a , z ) = D a 1 2 ( z ) . …
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 a , b , and x , 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
(For other notation see Notation for the Special Functions.)
k , m , n nonnegative integers.
primes on function symbols: derivatives with respect to argument.
The main function treated in this chapter is the Riemann zeta function ζ ( s ) . … The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .