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 unique solution with 
to the differential
equation
 |
(6)
|
(McCabe 1974).
Its Maclaurin series is given by
 |  |  |
(7)
|
 |  |  |
(8)
|
(OEIS A122803 and A001147). As 
along the real axis, it has the asymptotic
series
 |
(9)
|
For 
, it also arises in the Riemann-Liouville semi-integral
of 
with lower endpoint 0 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.
On the real axis, 
has its maximum at 
, or
 |
(15)
|
giving
 |
(16)
|
(OEIS A133841 and A133842), and its positive inflection
point 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, H. G. "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.
Comput. 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. L. III.
"On the Line-Absorption Coefficient Due to Doppler Effect and Damping."
Astrophys. J. 108, 112-115, 1948. 
." 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