Mathematical function
Plot of the Dawson integral function F(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D In mathematics , the Dawson function or Dawson integral [ 1] (named after H. G. Dawson [ 2] ) is the one-sided Fourier–Laplace sine transform of the Gaussian function.
The Dawson function, F ( x ) = D + ( x ) , {\displaystyle F(x)=D_{+}(x),} around the origin The Dawson function, D − ( x ) , {\displaystyle D_{-}(x),} around the origin The Dawson function is defined as either: D + ( x ) = e − x 2 ∫ 0 x e t 2 d t , {\displaystyle D_{+}(x)=e^{-x^{2}}\int _{0}^{x}e^{t^{2}}\,dt,} also denoted as F ( x ) {\displaystyle F(x)} or D ( x ) , {\displaystyle D(x),} or alternatively D − ( x ) = e x 2 ∫ 0 x e − t 2 d t . {\displaystyle D_{-}(x)=e^{x^{2}}\int _{0}^{x}e^{-t^{2}}\,dt.\!}
The Dawson function is the one-sided Fourier–Laplace sine transform of the Gaussian function , D + ( x ) = 1 2 ∫ 0 ∞ e − t 2 / 4 sin ( x t ) d t . {\displaystyle D_{+}(x)={\frac {1}{2}}\int _{0}^{\infty }e^{-t^{2}/4}\,\sin(xt)\,dt.}
It is closely related to the error function erf, as
D + ( x ) = π 2 e − x 2 erfi ( x ) = − i π 2 e − x 2 erf ( i x ) {\displaystyle D_{+}(x)={{\sqrt {\pi }} \over 2}e^{-x^{2}}\operatorname {erfi} (x)=-{i{\sqrt {\pi }} \over 2}e^{-x^{2}}\operatorname {erf} (ix)} where erfi is the imaginary error function, erfi(x ) = −i erf(ix ). Similarly, D − ( x ) = π 2 e x 2 erf ( x ) {\displaystyle D_{-}(x)={\frac {\sqrt {\pi }}{2}}e^{x^{2}}\operatorname {erf} (x)} in terms of the real error function, erf.
In terms of either erfi or the Faddeeva function w ( z ) , {\displaystyle w(z),} the Dawson function can be extended to the entire complex plane :[ 3] F ( z ) = π 2 e − z 2 erfi ( z ) = i π 2 [ e − z 2 − w ( z ) ] , {\displaystyle F(z)={{\sqrt {\pi }} \over 2}e^{-z^{2}}\operatorname {erfi} (z)={\frac {i{\sqrt {\pi }}}{2}}\left[e^{-z^{2}}-w(z)\right],} which simplifies to D + ( x ) = F ( x ) = π 2 Im [ w ( x ) ] {\displaystyle D_{+}(x)=F(x)={\frac {\sqrt {\pi }}{2}}\operatorname {Im} [w(x)]} D − ( x ) = i F ( − i x ) = − π 2 [ e x 2 − w ( − i x ) ] {\displaystyle D_{-}(x)=iF(-ix)=-{\frac {\sqrt {\pi }}{2}}\left[e^{x^{2}}-w(-ix)\right]} for real x . {\displaystyle x.}
For | x | {\displaystyle |x|} near zero, F (x ) ≈ x . For | x | {\displaystyle |x|} large, F (x ) ≈ 1/(2x ). More specifically, near the origin it has the series expansion F ( x ) = ∑ k = 0 ∞ ( − 1 ) k 2 k ( 2 k + 1 ) ! ! x 2 k + 1 = x − 2 3 x 3 + 4 15 x 5 − ⋯ , {\displaystyle F(x)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}\,2^{k}}{(2k+1)!!}}\,x^{2k+1}=x-{\frac {2}{3}}x^{3}+{\frac {4}{15}}x^{5}-\cdots ,} while for large x {\displaystyle x} it has the asymptotic expansion F ( x ) = 1 2 x + 1 4 x 3 + 3 8 x 5 + ⋯ . {\displaystyle F(x)={\frac {1}{2x}}+{\frac {1}{4x^{3}}}+{\frac {3}{8x^{5}}}+\cdots .}
More precisely | F ( x ) − ∑ k = 0 N ( 2 k − 1 ) ! ! 2 k + 1 x 2 k + 1 | ≤ C N x 2 N + 3 . {\displaystyle \left|F(x)-\sum _{k=0}^{N}{\frac {(2k-1)!!}{2^{k+1}x^{2k+1}}}\right|\leq {\frac {C_{N}}{x^{2N+3}}}.} where n ! ! {\displaystyle n!!} is the double factorial .
F ( x ) {\displaystyle F(x)} satisfies the differential equation d F d x + 2 x F = 1 {\displaystyle {\frac {dF}{dx}}+2xF=1\,\!} with the initial condition F ( 0 ) = 0. {\displaystyle F(0)=0.} Consequently, it has extrema for F ( x ) = 1 2 x , {\displaystyle F(x)={\frac {1}{2x}},} resulting in x = ±0.92413887... (OEIS : A133841 ), F (x ) = ±0.54104422... (OEIS : A133842 ).
Inflection points follow for F ( x ) = x 2 x 2 − 1 , {\displaystyle F(x)={\frac {x}{2x^{2}-1}},} resulting in x = ±1.50197526... (OEIS : A133843 ), F (x ) = ±0.42768661... (OEIS : A245262 ). (Apart from the trivial inflection point at x = 0 , {\displaystyle x=0,} F ( x ) = 0. {\displaystyle F(x)=0.} )
The Hilbert transform of the Gaussian is defined as H ( y ) = π − 1 P . V . ∫ − ∞ ∞ e − x 2 y − x d x {\displaystyle H(y)=\pi ^{-1}\operatorname {P.V.} \int _{-\infty }^{\infty }{\frac {e^{-x^{2}}}{y-x}}\,dx}
P.V. denotes the Cauchy principal value , and we restrict ourselves to real y . {\displaystyle y.} H ( y ) {\displaystyle H(y)} can be related to the Dawson function as follows. Inside a principal value integral, we can treat 1 / u {\displaystyle 1/u} as a generalized function or distribution, and use the Fourier representation 1 u = ∫ 0 ∞ d k sin k u = ∫ 0 ∞ d k Im e i k u . {\displaystyle {1 \over u}=\int _{0}^{\infty }dk\,\sin ku=\int _{0}^{\infty }dk\,\operatorname {Im} e^{iku}.}
With 1 / u = 1 / ( y − x ) , {\displaystyle 1/u=1/(y-x),} we use the exponential representation of sin ( k u ) {\displaystyle \sin(ku)} and complete the square with respect to x {\displaystyle x} to find π H ( y ) = Im ∫ 0 ∞ d k exp [ − k 2 / 4 + i k y ] ∫ − ∞ ∞ d x exp [ − ( x + i k / 2 ) 2 ] . {\displaystyle \pi H(y)=\operatorname {Im} \int _{0}^{\infty }dk\,\exp[-k^{2}/4+iky]\int _{-\infty }^{\infty }dx\,\exp[-(x+ik/2)^{2}].}
We can shift the integral over x {\displaystyle x} to the real axis, and it gives π 1 / 2 . {\displaystyle \pi ^{1/2}.} Thus π 1 / 2 H ( y ) = Im ∫ 0 ∞ d k exp [ − k 2 / 4 + i k y ] . {\displaystyle \pi ^{1/2}H(y)=\operatorname {Im} \int _{0}^{\infty }dk\,\exp[-k^{2}/4+iky].}
We complete the square with respect to k {\displaystyle k} and obtain π 1 / 2 H ( y ) = e − y 2 Im ∫ 0 ∞ d k exp [ − ( k / 2 − i y ) 2 ] . {\displaystyle \pi ^{1/2}H(y)=e^{-y^{2}}\operatorname {Im} \int _{0}^{\infty }dk\,\exp[-(k/2-iy)^{2}].}
We change variables to u = i k / 2 + y : {\displaystyle u=ik/2+y:} π 1 / 2 H ( y ) = − 2 e − y 2 Im i ∫ y i ∞ + y d u e u 2 . {\displaystyle \pi ^{1/2}H(y)=-2e^{-y^{2}}\operatorname {Im} i\int _{y}^{i\infty +y}du\ e^{u^{2}}.}
The integral can be performed as a contour integral around a rectangle in the complex plane. Taking the imaginary part of the result gives H ( y ) = 2 π − 1 / 2 F ( y ) {\displaystyle H(y)=2\pi ^{-1/2}F(y)} where F ( y ) {\displaystyle F(y)} is the Dawson function as defined above.
The Hilbert transform of x 2 n e − x 2 {\displaystyle x^{2n}e^{-x^{2}}} is also related to the Dawson function. We see this with the technique of differentiating inside the integral sign. Let H n = π − 1 P . V . ∫ − ∞ ∞ x 2 n e − x 2 y − x d x . {\displaystyle H_{n}=\pi ^{-1}\operatorname {P.V.} \int _{-\infty }^{\infty }{\frac {x^{2n}e^{-x^{2}}}{y-x}}\,dx.}
Introduce H a = π − 1 P . V . ∫ − ∞ ∞ e − a x 2 y − x d x . {\displaystyle H_{a}=\pi ^{-1}\operatorname {P.V.} \int _{-\infty }^{\infty }{e^{-ax^{2}} \over y-x}\,dx.}
The n {\displaystyle n} th derivative is ∂ n H a ∂ a n = ( − 1 ) n π − 1 P . V . ∫ − ∞ ∞ x 2 n e − a x 2 y − x d x . {\displaystyle {\partial ^{n}H_{a} \over \partial a^{n}}=(-1)^{n}\pi ^{-1}\operatorname {P.V.} \int _{-\infty }^{\infty }{\frac {x^{2n}e^{-ax^{2}}}{y-x}}\,dx.}
We thus find H n = ( − 1 ) n ∂ n H a ∂ a n | a = 1 . {\displaystyle \left.H_{n}=(-1)^{n}{\frac {\partial ^{n}H_{a}}{\partial a^{n}}}\right|_{a=1}.}
The derivatives are performed first, then the result evaluated at a = 1. {\displaystyle a=1.} A change of variable also gives H a = 2 π − 1 / 2 F ( y a ) . {\displaystyle H_{a}=2\pi ^{-1/2}F(y{\sqrt {a}}).} Since F ′ ( y ) = 1 − 2 y F ( y ) , {\displaystyle F'(y)=1-2yF(y),} we can write H n = P 1 ( y ) + P 2 ( y ) F ( y ) {\displaystyle H_{n}=P_{1}(y)+P_{2}(y)F(y)} where P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} are polynomials. For example, H 1 = − π − 1 / 2 y + 2 π − 1 / 2 y 2 F ( y ) . {\displaystyle H_{1}=-\pi ^{-1/2}y+2\pi ^{-1/2}y^{2}F(y).} Alternatively, H n {\displaystyle H_{n}} can be calculated using the recurrence relation (for n ≥ 0 {\displaystyle n\geq 0} ) H n + 1 ( y ) = y 2 H n ( y ) − ( 2 n − 1 ) ! ! π 2 n y . {\displaystyle H_{n+1}(y)=y^{2}H_{n}(y)-{\frac {(2n-1)!!}{{\sqrt {\pi }}2^{n}}}y.}
^ Temme, N. M. (2010), "Error Functions, Dawson's and Fresnel Integrals" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN 978-0-521-19225-5 , MR 2723248 . ^ Dawson, H. G. (1897). "On the Numerical Value of ∫ 0 h exp ( x 2 ) d x {\displaystyle \textstyle \int _{0}^{h}\exp(x^{2})\,dx} " . Proceedings of the London Mathematical Society . s1-29 (1): 519–522. doi :10.1112/plms/s1-29.1.519 . ^ Mofreh R. Zaghloul and Ahmed N. Ali, "Algorithm 916: Computing the Faddeyeva and Voigt Functions ," ACM Trans. Math. Soft. 38 (2), 15 (2011). Preprint available at arXiv:1106.0151 .