# Integral of Cauchy / Lorentzian distribution in Wolfram|Alpha

Posted 9 months ago
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 Hi,I am looking to do some line fitting of Lorentzian/Cauchy functions to a type of chromatogram. It is working fine, but I need an analytical expression to the area of the curve. The function is given as:f(x) = I*y^2/((x-x0)^2+y^2)where I,y,x0 are constants. Using Wolfram|Alpha to integrate, I get the following: F(x) = Iyatan((x-x0)/y) + CThe integral should be taken from -inf to inf. As atan approaches a constant value, the integral equals 0. Where am I going wrong in this? Is the integral undefined for this problem, or are my math skills just lacking? The integral should be dependent of I and y, but not x0.
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Posted 9 months ago
 I'm going to assume that I is not the imaginary number but a positive real constant. Using Mathematica I get the following (after changing I to i which might be the issue with Wolfram|Alpha): Integrate[i*y^2/((x - x0)^2 + y^2), {x, -\[Infinity], \[Infinity]}, Assumptions -> {x0 \[Element] Reals, y > 0, i > 0}] (* i \[Pi] y *) 
Posted 9 months ago
 Thanks for the reply! I found similar last night, when setting y=1 and I=1 and taking the numerical integral, thus getting pi, and then doubling y / I. Now I also realized my mistake... Missing a minus sign from the result from Wolfram|Alpha.F(x) = Iyatan((x-x0)/y) + C In interval -inf...inf A = Iy(atan(inf)-atan(-inf)) A = Iypi