Gaussian Integral

I=letex2dx=ey2dyI\overset { let }{ = } \int _{ -\infty }^{ \infty }{ { e }^{ -{ x }^{ 2 } }dx } =\int _{ -\infty }^{ \infty }{ { e }^{ -{ y }^{ 2 } }dy }

I2=(I)(I)=(ex2dx)(ey2dy)=e(x2+y2)dxdy=02π0rer2drdθ=02π[er22]0dθ=02π12dθ=[θ2]02πI2=πI=π(I>0)\begin{aligned} { I }^{ 2 } & = & (I)(I) \\ \quad & = & (\int _{ -\infty }^{ \infty }{ { e }^{ -{ x }^{ 2 } }dx } )(\int _{ -\infty }^{ \infty }{ { e }^{ -{ y }^{ 2 } }dy } ) \\ \quad & = & \int _{ -\infty }^{ \infty }{ \int _{ -\infty }^{ \infty }{ { e }^{ -{ (x }^{ 2 }+{ y }^{ 2 }) }dx } dy } \\ \quad & = & \int _{ 0 }^{ 2\pi }{ \int _{ 0 }^{ \infty }{ r{ e }^{ -{ r }^{ 2 } }dr } d\theta } \\ \quad & = & \int _{ 0 }^{ 2\pi }{ { \left[ -\frac { { e }^{ -{ r }^{ 2 } } }{ 2 } \right] }_{ 0 }^{ \infty }d\theta } \\ \quad & = & \int _{ 0 }^{ 2\pi }{ \frac { 1 }{ 2 } d\theta } \\ \quad & = & { \left[ \frac { \theta }{ 2 } \right] }_{ 0 }^{ 2\pi } \\ { I }^{ 2 } & = & \pi \\ I & = & \sqrt { \pi } \quad (I>0) \end{aligned}

ex2dx=π\boxed { \displaystyle \int _{ -\infty }^{ \infty }{ { e }^{ -{ x }^{ 2 } }dx } =\sqrt { \pi } }

Note by Gordon Chan
1 week, 5 days ago

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