 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2021-resit/1/b/solution

The fundamental theorem of calculus differentiates the variable upper endpoint:

I^{'} (x) = \frac{2}{π} e^{- x^{2}}, I^{''} (x) = - \frac{4 x}{π} e^{- x^{2}}, I^{'''} (x) = \frac{- 4 + 8 x ^{2}}{π} e^{- x^{2}} .

At zero,

I (0) = 0

I^{'} (0) = 2/ π

I^{''} (0) = 0

and

I^{'''} (0) = - 4/ π

. The integrand is even, so its integral from zero is an odd function; all even coefficients in the Maclaurin series vanish. Hence

I (x) = \frac{2}{π} (x - \frac{x ^{3}}{3}) + O (x^{5}) .

This is the local expansion of the error function.

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