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In this problem, we will investigate gaussian integrals and theFeynman technique to study differentiation under the integralsymbol. We will start with the integral of the Gaussian.
In this problem we will investigate about gaussian integrals and the Feynnman technique to study differentiation under the integral symbol. We will start with the integral of the Gaussian I=∫−∞∞​e−x2dx (a) take x=rcosθ and y=rsinθ and find ∂(r,θ)∂(x,r)​. (b) What will be the expression for dA=dxdy in terms of r and θ (c) Consider the integral in terms of y to show that I=∫−∞∞​e−y2dy (d) Now combine equation 1 and 2 to show that I2=∫−∞∞​∫−∞∞​e−(x2+y2)dxdy (e) Convert equation 3 in terms of r and θ to show that I2=∫02π​∫0∞​re−r2drdθ (f) Evaluate equation 4 to show that ∫−∞∞​e−x2dx=π​ 2 (g) Using the results of the previous part, show that ∫−∞∞​e−αx2dx=απ​​ (h) Differentiate both sides of 6 with respect to α to evaluate ∫−∞∞​x2e−αx2dx (i) Now finally differentiate the equation in 62n times to show that ∫−∞∞​x2ne−αx2dx=23nn!(2n+1)!​