Integral from a function that has a singularity

2017 1 29
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Hello I want to take an integral from the function that has a singularity(pole), by quadgk but this common don't give a right answer .for instance an integral of the function 1/(x-1) from(-2,2) anyone can guide me thanks

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Star Strider
Star Strider 2017-1-29

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The best I can do is to rely on the Symbolic Math Toolbox, that came up with a piecewise result. Perhaps you can use this in your code:
syms x L U
f(x) = 1/(x-1);
intf = int(f,x, L, U)
intf =
piecewise(L <= 1 & 1 <= U, int(1/(x - 1), x, L, U), 1 < L | U < 1, log(U - 1) - log(L - 1))
The ‘L’ and ‘U’ variables are the lower and upper limits of integration, respectively.

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Kurt Hakan
Kurt Hakan 2017-1-29
thanks for your answer. I would like to ask my question from other point of view. MATLAB Saied in help, the quadgk common can solve the integral of functions that have a singularity but here cant solve the problem.??!!
Star Strider
Star Strider 2017-1-29
My pleasure.
In R2016b, the quadgk function will do the integration without error, but will throw Warnings. The quadgk function will also accept a 'Waypoints' argument, where you can ‘inform’ it about the singularities that exist within the limits of integration, and integrate around them if you wish.
For example:
f = @(x) 1./(x-1);
int1 = quadgk(f, -2, 2)
int2 = quadgk(f, -2, 2, 'Waypoints',[1, 1], 'MaxIntervalCount',1E+7)
int1 =
12.9845608756444
int2 =
-0.926665144495417
They both throw warnings. You can also divide the integral into two regions, from -2 to 1 and 1 to +2 and add them.
I am not certain there are any good ways to deal with such problems, although there are applied mathematicians here who I hope will add to this discussion.
Kurt Hakan
Kurt Hakan 2017-1-29
编辑：Kurt Hakan 2017-1-29
int2 = -0.926665144495417 is not correct answer and in we divide the(-2,2) to (-2,1)and (1,2) and take an integral the answer is
int3 =
-1.1859
but the correct answer is (-log(3)=-1.09861).
???????
Star Strider
Star Strider 2017-1-29
If you specify the limits of integration to go from -2 to +2 with 'Waypoints' around the singularity (doing complex contour integration), you get exactly that result!
int4 = quadgk(f, -2, 2, 'Waypoints',[1+1i, 1-1i], 'MaxIntervalCount',1E+7)
int4 =
-1.09861228866811
Star Strider
Star Strider 2017-1-29
As always, my pleasure.
Wolfram Alpha evaluates it using the Cauchy principal value

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