反常积分∫0到无穷e^(-x^2)dx=
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反常积分∫0到无穷e^(-x^2)dx=
反常积分∫0到无穷e^(-x^2)dx=
反常积分∫0到无穷e^(-x^2)dx=
二重积分的极坐标变换
∫e^(-x²)dx=∫e^(-y²)dy
故(∫e^(-x²)dx)²
=∫e^(-x²)dx∫e^(-y²)dy
=∫∫e^[-(x²+y²)]dxdy
=∫dθ∫e^(-r²)rdr
=2π∫e^(-r²)rdr
=-π∫e^(-r²)d(-r²)
=-πe^(-r²)|
=π
即∫e^(-x²)dx=√π
k1 = ∫0到无穷e^(-x^2)dx
k2 = ∫0到无穷e^(-y^2)dy
k1*k2 =
∫0到无穷
∫0到无穷e^(-x^2)dx e^(-y^2)dy = ∫0到无穷 ∫0到无穷 e^[(-x^2)+(-y^2)dx dy
转到极坐标:
x^2 + y^2 = r^2 ; dxdy = r dr d(theta)
积...
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k1 = ∫0到无穷e^(-x^2)dx
k2 = ∫0到无穷e^(-y^2)dy
k1*k2 =
∫0到无穷
∫0到无穷e^(-x^2)dx e^(-y^2)dy = ∫0到无穷 ∫0到无穷 e^[(-x^2)+(-y^2)dx dy
转到极坐标:
x^2 + y^2 = r^2 ; dxdy = r dr d(theta)
积分是在第一象限:
k1*k2 =
∫ 0到pi/2 [ ∫0到无穷 e^(-r^2)rdr ] d(theta)
=
∫ 0到pi/2 [(1/2) ∫0到无穷 e^(-r^2)d(r^2) ] d(theta)
let z=r^2,
k1*k2 =
∫ 0到pi/2 [(1/2) ∫0到无穷 e^(-z)dz ] d(theta) =
∫ 0到pi/2 (1/2) d(theta) = (1/2)*(pi/2)
= pi/4
so k1 = (pi/4)^(0.5)
收起