residue complex number by matlab

how can I find the residue of (z+1)/(z^3 (z-2)) by using matlab ?

Answer 1

[John D'Errico]

I can't recall if I ever published this tool. I thought I did.

fun = @(z) (z+1)./(z.^3.*(z-2));
format long g

We know z=2 to be a first order pole.

[res,err] = residueEst(fun,2,'poleorder',1)
res =
                   0.375
err =
     8.8624722305483e-16

And z=0 is a 3rd order pole.

[res,err] = residueEst(fun,0,'poleorder',3)
res =
      -0.187500006276744
err =
    2.93992309142077e-09

Expect a wee bit less accuracy around higher order poles, but it still did reasonably well.

As I said, I thought it was posted on the File Exchange, but it may not have been. I've written so many neat toys over the years, that sometimes I forget to post them...

As it turns out, I never did post it. I'll update the limest submission to include residueEst, as they are both limit calculators.

[adrian zizo]

it doesn't worke .. my matlab version is R2010a

[John D'Errico]

I've just posted residueEst as a part of my LIMEST File Exchange submission. It includes demos of use.

[John D'Errico]

I wrote it in 2008, and it has not changed since then. It WILL work. But you need to download it from the file exchange. That tool is not part of MATLAB. You cannot hope it will work unless you get the file.

[adrian zizo]

I have another problem Mr.John D'Errico .. how can I found the value of complex contour integral : example ∮z/((z^2+4)(z-1)) dz around z-2=2 (center of circle is 2 and radius equal 2)

I answer it by using matlab but it give me error ...

this is my answer :

>> fun=@(z) (z)./((z.^2+4)*(z-1));

>> g=@(theta) (2+2*cos(theta))+2i*sin(theta);

>> gprime=@(theta)-2*sin(theta)+2i*cos(theta);

>> q1=quad(@(t) fun(g(t)).*gprime(t),0,2*pi)

??? Error using ==> mtimes

Inner matrix dimensions must agree.

Error in ==> @(z)(z)./((z.^2+4)*(z-1))

Error in ==> @(t)fun(g(t)).*gprime(t)

Error in ==> quad at 77

y = f(x, varargin{:});

[John D'Errico]

Ok, so you know how to use .^ and ./ to solve some problems. But is there a reason why you did not use .* also?

fun=@(z) (z)./((z.^2+4)*(z-1));  % your function
fun=@(z) (z)./((z.^2+4).*(z-1)); % my version

See that the error came from mtimes.

[Mike Hosea]

FYI, in case it comes in handy at some point, the INTEGRAL function (and QUADGK) can do contour integrals over piecewise linear paths in the complex plane using the 'Waypoints' option. For example, integrating over square path enclosing z = 2

>> integral(f,1,1,'waypoints',[1-1i,3-1i,3+1i,1+1i])
ans =
                          0 +      2.35619449019235i
>> ans/(2*pi*1i)
ans =
                     0.375

[John D'Errico]

That is good to know. Too often these tools contain nice additional abilities that we old-timers never realize were added.

[Niklas Kurz]

what if there is an indetermined variable in the function? (fun = @(z,a)). Can it deal with two parameters?