Approximation of the error function

Calculus Level 5

As it turns out, a good approximation for the error function erf ( x ) \textrm{erf}(x) is the function tanh ( 2 x π ) \tanh\left(\dfrac{2x}{\sqrt{\pi}}\right) .

How good of an approximation? Tell me yourself by finding the total area bounded by these two functions. That is, find the value of:

erf ( x ) tanh ( 2 x π ) d x \int_{-\infty}^{\infty} \left|\textrm{erf}(x) -\tanh\left(\frac{2x}{\sqrt{\pi}}\right)\right| dx

Please round to 3 decimal places.

Note: erf ( x ) = 2 π 0 x e t 2 d t \displaystyle \textrm{erf}(x)= \dfrac{2}{\sqrt{\pi}}\int_{0}^{x} e^{-t^2} dt and tanh ( x ) = sinh ( x ) cosh ( x ) \tanh(x)= \dfrac{\sinh(x)}{\cosh(x)} .


The answer is 0.1.

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1 solution

Andreas Wendler
Mar 3, 2016

Solved it via X(PLORE):

f(x)=in(2/sqrt(pi)*exp(-t^2),t=0 to x)

g(x)=tanh(2*x/sqrt(pi))

in(abs(f(x)-g(x)),x=-10 to 10)

So I got 0.100192 ± 5.4e-11.

Rem.: Deviations for larger integration borders are negligible with respect to the necessary decimal places.

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