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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.19 Voigt Functions

Contents
  1. §7.19(i) Definitions
  2. §7.19(ii) Graphics
  3. §7.19(iii) Properties
  4. §7.19(iv) Other Integral Representations

§7.19(i) Definitions

For x and t>0,

7.19.1 U(x,t)=14πte(xy)2/(4t)1+y2dy,
7.19.2 V(x,t)=14πtye(xy)2/(4t)1+y2dy.
7.19.3 U(x,t)+iV(x,t)=π4tez2erfcz,
z=(1ix)/(2t).
7.19.4 H(a,u)=aπet2dt(ut)2+a2=1aπU(ua,14a2).

H(a,u) is sometimes called the line broadening function; see, for example, Finn and Mugglestone (1965).

§7.19(ii) Graphics

See accompanying text
Figure 7.19.1: Voigt function U(x,t), t=0.1, 2.5, 5, 10. Magnify
See accompanying text
Figure 7.19.2: Voigt function V(x,t), t=0.1, 2.5, 5, 10. Magnify

§7.19(iii) Properties

7.19.5 limt0U(x,t) =11+x2,
limt0V(x,t) =x1+x2.
7.19.6 U(x,t) =U(x,t),
V(x,t) =V(x,t).
7.19.7 0 <U(x,t)1,
1 V(x,t)1.
7.19.8 V(x,t) =xU(x,t)+2tU(x,t)x,
7.19.9 U(x,t) =1xV(x,t)2tV(x,t)x.

§7.19(iv) Other Integral Representations

7.19.10 U(ua,14a2)=a0eat14t2cos(ut)dt,
7.19.11 V(ua,14a2)=a0eat14t2sin(ut)dt.