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单词 GeneralisationOfGaussianIntegral
释义

generalisation of Gaussian integral


The integral

0e-x2costxdx:=w(t)

is a generalisation of the Gaussian integral  w(0)=π2.  For evaluating it we first form its derivativePlanetmathPlanetmath which may be done by differentiating under the integral sign (http://planetmath.org/DifferentiationUnderIntegralSign):

w(t)=0e-x2(-x)sintxdx=120e-x2(-2x)sintxdx

Using integration by parts this yields

w(t)=12/x=0e-x2sintx-t20e-x2costxdx=12(0-0)-t20e-x2costxdx=-t2w(t).

Thus w(t) satisfies the linear differential equation

dwdt=-12tw,

where one can separate the variables (http://planetmath.org/SeparationOfVariables) and integrate:

dww=-12t𝑑t.

So,  lnw=-14t2+lnC,  i.e.  w=w(t)=Ce-14t2, and since there is the initial conditionMathworldPlanetmathw(0)=π2, we obtain the result

w(t)=π2e-14t2.
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更新时间:2025/5/4 19:59:19