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Erfc Differential Equation

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The second-order ordinary differential equation

y^('')+2xy^'-2ny=0,
(1)

whose solutions may be written either

y=Aerfc_n(x)+Berfc_n(-x),
(2)

where erfc_n(x) is the repeated integral of the erfc function (Abramowitz and Stegun 1972, p. 299), or

y=C_1e^(-x^2)H_(-n-1)(x)+C_2e^(-x^2)_1F_1(1/2(n+1);1/2;x^2),
(3)

where H_n(x) is a Hermite polynomial and _1F_1(a;b;z) is a confluent hypergeometric function of the first kind.

See also

Erfc

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 299, 1972.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997.

Referenced on Wolfram|Alpha

Erfc Differential Equation

Cite this as:

Weisstein, Eric W. "Erfc Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ErfcDifferentialEquation.html

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