Gaussian Integral

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

\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}

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

Note by Gandoff Tan
9 months, 4 weeks ago

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Ajuba is a member of LIM domain containing proteins that contribute to cell fate determination and regulate cell proliferation and differentiation. Ajuba is also involved in the regulation of actin cytoskeleton dynamics and cell migration. Ajuba plays a important role in the regulation of the kinase activity of AURKA/Aurora A for mitotic commitment.

https://www.creative-biogene.com/genesearch/Ajuba.html

- 8 months ago