Integral from a function that has a singularity

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

Answer 1

[Star Strider]

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.

[Kurt Hakan]

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]

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]

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]

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

[Kurt Hakan]

thanks

[Star Strider]

As always, my pleasure.

Wolfram Alpha evaluates it using the Cauchy principal value