# error function

(redirected from Complementary error function)
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## error function

[′er·ər ‚fəŋk·shən]
(mathematics)
The real function defined as the integral from 0 to x of e -t 2 dt or e t 2 dt, or the integral from x to ∞ of e -t 2 dt.
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Double integrals are reduced into line integrals in terms of complementary error functions. For each linear integral, the numerical SDP method and Stokes lines are used to achieve the frequency independency in computing the highly oscillatory line integral.
After integration by parts, it has the closed form formula related to special complementary error functions erfc (z) as follows:
The two dimensional integral is reduced to line integrals related to complementary error functions. Steepest descent paths deformation on the complex plane are used to calculate the highly oscillatory line integrals.
We solve this equation numerically by the method of bisections, for x = 5(0.5) 100 and d = 0, c = 2; d = 0, c = 5; and d = 1(1) 10, c = 1, using rational approximations of Cody  to evaluate the complementary error function.
where erfc{*} denotes the complementary error function.
According to Ref., the complementary error function erfc(x) can be approximately expressed as erfc(x) [congruent to] [[summation].sup.2.sub.m=1][u.sub.m] exp(-[v.sub.m][x.sup.2]), where {[u.sub.m]} = {1/6,1/2}, {[v.sub.m]} = {1,4/3}.

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