?The Error Function The error function\(\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}

Chapter 4, Problem 80

(choose chapter or problem)

The Error Function The error function

\(\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t\)

is used in probability, statistics, and engineering.
(a) Show that \(\int_{a}^{b} e^{-t^{2}} d t=\frac{1}{2} \sqrt{\pi}[\operatorname{erf}(b)-\operatorname{erf}(a)]\)
(b) Show that the function \(y=e^{x^{2}} \operatorname{erf}(x)\)satisfies the differential equation \(y^{\prime}=2 x y+2 / \sqrt{\pi}\).

Equation Transcription:

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Text Transcription:

erf(x)=2/sqrt pi integral _0 ^x e^-t^2 dt

integral _a ^b e^-t^2 dt=1/2 sqrt pi [erf(b)-erf(a)]

y=e^x^2 erf(x)

y prime=2xy+2/sqrt pi