Содержание

What Every Young Mathematician Should Know

Some Interesting Applications of Integral Calculus

One Interesting Integral

The int{-infty}{+infty}{e^{x^2}} dx integral.

Here we evaluate an interesting definite integral of which many young mathematicians seem to be ignorant.

Theorem1(Euler–Poisson) The value of int{-infty}{+infty}{e^{x^2}}dx is int{-infty}{+infty}{e^{x^2}}dx=sqrt{pi}

Proof: We have delim{(}{(int{-infty}{+infty}{e^{x^2}}dx)}{)}^2=delim{(}{(int{-infty}{+infty}{e^{x^2}}dx)}{)}~delim{(}{(int{-infty}{+infty}{e^{x^2}}dx)}{)},

which by the Fubini theorem can be represented as follows:

doubleint{-infty}{+infty}{e^{x^2}e^{y^2}dxdy}=doubleint{-infty}{+infty}{e^{x^2+y^2}dxdy}.

Let us implement the change of variables in order to have the polar coordinates:

delim{lbrace}{matrix{3}{1}{{x = r cos theta} {y = r sin theta}}}{}.

One can easily see that the Jacobian determinant of this transformation is r, so then we have

int{0}{2pi}{int{0}{+infty}{e^{-r^2}r~dr~d theta}}=int{0}{2pi}{delim{[}{int{0}{+infty}{e^{-r^2}r~dr~}{]}d theta}}=pi

Remark. A mathematician is one to whom that is as obvious as that twice two makes four is to you.


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