Gaussian Integral

\int_{- \infty}^{\infty} e^{- x^{2}} d x = π

I = \int_{- \infty}^{\infty} e^{- \frac{1}{2} a x^{2} + b x} d x = \frac{2 π}{a} exp (\frac{b ^{2}}{2 a})

\frac{\partial I}{\partial b} = \int_{- \infty}^{\infty} x e^{- \frac{1}{2} a x^{2} + b x} d x = \frac{b}{a} \frac{2 π}{a} exp (\frac{b ^{2}}{2 a})

\frac{\partial ^{2} I}{\partial b ^{2}} = \int_{- \infty}^{\infty} x^{2} e^{- \frac{1}{2} a x^{2} + b x} d x = \frac{1}{a} (1 + \frac{b ^{2}}{a}) \frac{2 π}{a} exp (\frac{b ^{2}}{2 a})

Lin's Notes Garden