# taylor expansion of erf(x)

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• January 10th 2010, 08:27 AM
alexandrabel90
taylor expansion of erf(x)
how do you solve:

consider the so called error integral erf(x) for x ∈R:
erf(x)= 2/ (pi)^(1/2) integrate exp (-y) from x-x to x=0.
derive the first few terms in taylor expansion of erf(x) up to x^3. hint: use second fundamental theorem of calculus

can someone help me solve this? im completely lost..
• January 10th 2010, 10:01 AM
skeeter
Quote:

Originally Posted by alexandrabel90
how do you solve:

consider the so called error integral erf(x) for x ∈R:
erf(x)= 2/ (pi)^(1/2) integrate exp (-y) from x-x to x=0. ???
derive the first few terms in taylor expansion of erf(x) up to x^3. hint: use second fundamental theorem of calculus

can someone help me solve this? im completely lost..

$erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt$

using the 2nd FTC ...

$erf'(x) = \frac{2}{\sqrt{\pi}} e^{-x^2}$

note that $e^u = 1 + u + \frac{u^2}{2!} + \frac{u^3}{3!} + ...$

so ...

$e^{-x^2} = 1 - x^2 + \frac{x^4}{2!} - \frac{x^6}{3!} + ...$

integrate term for term ...

$\frac{2}{\sqrt{\pi}} \int e^{-x^2} \, dx = \frac{2}{\sqrt{\pi}} \left(C + x - \frac{x^3}{3} + \frac{x^5}{5 \cdot 2!} - \frac{x^7}{7 \cdot 3!} + ...\right)$

finish up ...
• January 10th 2010, 10:52 AM
alexandrabel90
do i need to find the remainder since this is a taylor series question? or what else do i need to do to finish up?
• January 10th 2010, 11:26 AM
skeeter
Quote:

Originally Posted by alexandrabel90
do i need to find the remainder since this is a taylor series question? or what else do i need to do to finish up?

the directions state ...

Quote:

derive the first few terms in taylor expansion of erf(x) up to x^3