Goodwin–Staton integral

In mathematics, the Goodwin–Staton integral is the special function ^[1]

G(z)=\int _{0}^{\infty }{\frac {e^{-t^{2}}}{t+z}}\,dt,\qquad |\arg z|<\pi .

The restriction on the argument excludes the non-positive real axis, where the denominator can vanish along the path of integration. The integral is named after E. T. Goodwin and J. Staton, who published a numerical table of the corresponding real integral in 1948.^[2]

Relation to other special functions

For positive real $x$ , the Goodwin–Staton integral is related to Dawson's integral $F$ and the exponential integral by^[3]

G(x)={\sqrt {\pi }}\,F(x)-{\frac {1}{2}}e^{-x^{2}}\mathrm {Ei} (x^{2}),\qquad x>0.

Differential equations

Differentiating the defining integral and integrating by parts gives the first-order inhomogeneous differential equation

G'(z)+2zG(z)={\sqrt {\pi }}-{\frac {1}{z}}.

Differentiating this relation twice and eliminating the inhomogeneous term gives the third-order linear homogeneous equation

zG'''(z)+(2+2z^{2})G''(z)+8zG'(z)+4G(z)=0.

Properties on the positive real axis

For $x>0$ , differentiation under the integral sign gives

G^{(n)}(x)=(-1)^{n}n!\int _{0}^{\infty }{\frac {e^{-t^{2}}}{(t+x)^{n+1}}}\,dt.

Hence

(-1)^{n}G^{(n)}(x)>0,\qquad n=0,1,2,\ldots ,

so $G$ is a completely monotonic function on $(0,\infty )$ . In particular, $G$ is positive, strictly decreasing, and convex there.

Series near the origin

D. S. Jones studied the generalized Goodwin–Staton integral and obtained convergent series representations.^[4] Specializing Jones's equation (2.11) gives, for $|\arg z|<\pi$ ,

G(z)=e^{-z^{2}}\left(-\log z-{\frac {\gamma }{2}}-{\frac {1}{2}}\sum _{m=1}^{\infty }{\frac {z^{2m}}{m!\,m}}+{\frac {\sqrt {\pi }}{2}}\sum _{m=0}^{\infty }{\frac {z^{2m+1}}{m!\left(m+{\frac {1}{2}}\right)}}\right),

where $\log z$ is the principal logarithm and $\gamma$ is the Euler–Mascheroni constant.

Consequently, as $z\to 0$ within the cut plane,

G(z)=-\log z-{\frac {\gamma }{2}}+{\sqrt {\pi }}\,z+\left(\log z+{\frac {\gamma }{2}}-{\frac {1}{2}}\right)z^{2}-{\frac {2{\sqrt {\pi }}}{3}}z^{3}+\left({\frac {3}{8}}-{\frac {\gamma }{4}}-{\frac {1}{2}}\log z\right)z^{4}+O(z^{5}).

In particular,

G(x)\sim -\log x-{\frac {\gamma }{2}}\qquad (x\to 0^{+}).

Asymptotic expansion

For large positive $x$ , the integral has the asymptotic expansion^[4]

G(x)\sim {\frac {1}{2}}\sum _{n=0}^{\infty }(-1)^{n}\Gamma \left({\frac {n+1}{2}}\right)x^{-n-1}\qquad (x\to +\infty ).

Thus its first terms are

G(x)\sim {\frac {\sqrt {\pi }}{2x}}-{\frac {1}{2x^{2}}}+{\frac {\sqrt {\pi }}{4x^{3}}}-{\frac {1}{2x^{4}}}+{\frac {3{\sqrt {\pi }}}{8x^{5}}}-{\frac {1}{x^{6}}}+\cdots .

The NIST Digital Library of Mathematical Functions refers to Olver for corresponding expansions with bounds on the remainder for both real and complex values of the argument.^[5]

Generalized Goodwin–Staton integral

Jones considered the generalized integral^[4]

I(\mu ,z)=\int _{0}^{\infty }{\frac {t^{\mu }e^{-t^{2}}}{t-z}}\,dt,

where $\mu$ is real with $\mu >-1$ and $z$ lies off the positive real axis. The ordinary Goodwin–Staton integral is the special case

G(z)=I(0,-z).

The generalized integral is a Stieltjes transform and satisfies the recurrence

I(\mu +1,z)={\frac {1}{2}}\Gamma \left({\frac {\mu +1}{2}}\right)+zI(\mu ,z),

as well as

{\frac {\partial I(\mu ,z)}{\partial z}}=\mu I(\mu -1,z)-2I(\mu +1,z),\qquad \mu >0.

Jones derived convergent series, asymptotic expansions, and explicit error bounds, and also treated oscillatory and logarithmic variants.^[4] Mamedov later gave another evaluation of generalized Goodwin–Staton integrals using the binomial expansion theorem.^[6]

Numerical evaluation

For positive real arguments, numerical evaluation can be reduced to Dawson's integral and the exponential integral through the relation above.^[3] For small and moderate arguments, Jones's convergent series can be used, while for large arguments the asymptotic expansion is more suitable.^[4] The DLMF section on computation of the Goodwin–Staton integral cites the original table of Goodwin and Staton.^[7]

References

↑ "DLMF §7.2(v), Goodwin–Staton Integral". NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology. Retrieved 9 August 2026.
↑ Goodwin, E. T.; Staton, J. (1948). "Table of f(x)=∫₀^∞ e^(−u²)/(u+x) du". The Quarterly Journal of Mechanics and Applied Mathematics. 1 (1): 319–326. doi:10.1093/qjmam/1.1.319.
1 2 "DLMF §7.5, equation 7.5.13". NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology. Retrieved 9 August 2026.
1 2 3 4 5 Jones, D. S. (2005). "The generalized Goodwin–Staton integral". Proceedings of the Edinburgh Mathematical Society. 48 (3): 635–650. doi:10.1017/S0013091504001087.
↑ "DLMF §7.12(iii), Goodwin–Staton Integral". NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology. Retrieved 9 August 2026.
↑ Mamedov, B. A. (2007). "Evaluation of the generalized Goodwin–Staton integral using binomial expansion theorem". Journal of Quantitative Spectroscopy and Radiative Transfer. 105 (1): 8–11. doi:10.1016/j.jqsrt.2006.09.018.
↑ "DLMF §7.22(ii), Goodwin–Staton Integral". NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology. Retrieved 9 August 2026.