Gauss integrál

Tétel:

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

Biz.:

(\int_{- \infty}^{\infty} e^{- x^{2}} d x)^{2} = (\int_{- \infty}^{\infty} e^{- x^{2}} d x) \cdot (\int_{- \infty}^{\infty} e^{- y^{2}} d y) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{- x^{2} - y^{2}} d x d y

= \int_{0}^{2 π} \int_{0}^{\infty} e^{- r^{2}} r d r d θ = \int_{0}^{2 π} - \frac{1}{2} \int_{0}^{\infty} - 2 r e^{- r^{2}} d r d θ = \int_{0}^{2 π} - \frac{1}{2} [e^{- r^{2}}]_{0}^{\infty} d x

= - \frac{1}{2} [e^{- r^{2}}]_{0}^{\infty} \cdot 2 π = - \frac{lim _{r \to \infty} e ^{- r^{2}}}{2} 2 π + \frac{1}{2} 2 π = - \frac{0}{2} + π = π

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

Dependencies:

Jegyzetek kincstára