Dawson's integral (Abramowitz and Stegun 1972, pp. 295 and 319), also sometimes called Dawson's function, is the entire function given by the integral
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is erfi, that arises in computation of the Voigt lineshape
(Harris 1948, Hummer 1963, Sajo 1993, Lether 1997), in heat conduction, and in the
theory of electrical oscillations in certain special vacuum tubes (McCabe 1974).
It is commonly denoted 
(McCabe 1974; Coleman 1987; Milone and Milone 1988; Sajo 1993; Lether 1997; Press
et al. 2007, p. 302), although Spanier and Oldham (1987) denote it by
.
Dawson's integral is implemented in the Wolfram Language as DawsonF[z].
It is an odd function, so
 |
(3)
|
Its derivative is
 |
(4)
|
and its indefinite integral is
 |
(5)
|
where 
is a generalized hypergeometric
function.
It is the particular solution to the differential equation
 |
(6)
|
(McCabe 1974).
Its Maclaurin series is given by
 |  |  |
(7)
|
 |  |  |
(8)
|
(OEIS A122803 and A001147). If has the asymptotic series
 |
(9)
|
It also arises in the semi-integral of 
via
 |
(10)
|
(Spanier and Oldham 1987, p. 406).
It is given by the sums
 |  |  |
(11)
|
 |  |  |
(12)
|
(Spanier and Oldham 1987, p. 407), where 
is the gamma function
and 
is a Pochhammer symbol.
Dawson's integral has continued fractions
 |  |  |
(13)
|
 |  |  |
(14)
|
(McCabe 1974).
The plots above show the behavior of 
in the complex plane.
has a maximum at 
, or
 |
(15)
|
giving
 |
(16)
|
(OEIS A133841 and A133842), and an inflection at 
,
or
 |
(17)
|
giving
 |
(18)
|
(OEIS A133843).
The function is sometimes generalized such that
 |
(19)
|
giving
 |  |  |
(20)
|
 |  |  |
(21)
|
where 
is the erf function and 
is the imaginary error function erfi.
The plots above show the behavior of 
in the complex plane.
-Method: Dawson's Integral." J. Comput. Appl. Math. 20,
137-151, 1987.Dawson, F. "On the Numerical Value of 
." London Math. Soc. Proc. 29,
519-522, 1898.Dijkstra, D. A. "A Continued Fraction Expansion
for a Generalization of Dawson's Integral." Math. Comp. 31, 503-510,
1977.Faddeyeva, V. N. and Terent'ev, N. M. Tables of Values
of the Function 
for Complex Argument. New York: Pergamon Press, 1961.Harris, D.
III. "On the Line Absorption Coefficients Due to Doppler Effect and Damping."
Astrophys. J. 108, 1120-115, 1948.Hummer, D. G. "Noncoherent
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Not. Roy. Astron. Soc. 125, 21-37, 1963.Hummer, D. G.
"Expansion of Dawson's Function in a Series of Chebyshev Polynomials."
Math. Comput. 18, 317-319, 1964.Lether, F. G. "Elementary
Approximations for Dawson's Integral." J. Quant. Spectros. Radiat. Transfer 4,
343-345, 1991.Lether, F. G. "Constrained Near-Minimax Rational
Approximations to Dawson's Integral." Appl. Math. Comput. 88,
267-274, 1997.Lohmander, B. and Rittsten, S. "Table of the Function
." Kungl.
Fysiogr. Sällsk. i Lund Föhr. 28, 45-52, 1958.Luke,
Y. L. 
and 
."
Brooklyn, NY: Mapleton House, 1948.Rybicki, G. B. "Dawson's
Integral and the Sampling Theorem." Computers in Physics 3, 85-87,
1989.Sajo, E. "On the Recursive Properties of Dawson's Integral."
J. Phys. A 26, 2977-2987, 1993.Sloane, N. J. A.
Sequences