How to solve such integrations on MATLAB?

For domain, i.e, z is a complex variable in the unit disk, and for n any natural number, we can integrate functions like F(z) but MATLAB doesn't give answers to integrations like G(z), how should i do it? Also, my limits of integration are 0 to z where z .

Is there any numerical method that we can apply like quadgk?, you can see a reference of how to apply quadgk on such integrations in one of my questions asked because i didn't understand that. Also, some other analytic technique if there is like partial fractions or something?

syms z n;
F=int(1/(1-z^n),z)
G=int(z^(3*n-1)/(1-z^(2*n)),z)

Answer 1

[Torsten]

Invest a little time and deduce

z^(3*n-1)/(1-z^(2*n)) = -z^(n-1) + 0.5*z^(n-1)/(1-z^n) + 0.5*z^(n-1)/(1+z^n)

This function can be integrated analytically:

syms z
syms n integer
f = -z^(n-1) + 0.5*z^(n-1)/(1-z^n) + 0.5*z^(n-1)/(1+z^n);
F = int(f)

F = 

simplify(diff(F))

ans = 

Answer 2

[Mrutyunjaya Hiremath]

The 'quadgk' function in MATLAB can be used for numerical integration. However, since you're dealing with complex variables, you might need to handle the real and imaginary parts separately or look for specialized numerical methods for complex functions.

Here's how you can use quadgk to evaluate the integral for F(z):

n_value = 2; % replace with the value of n you're interested in
z_value = 0.5 + 0.5i; % replace with the value of z you're interested in

realPart_F = @(z) real(1 ./ (1 - z .^ n_value));
imagPart_F = @(z) imag(1 ./ (1 - z .^ n_value));

real_integral = quadgk(realPart_F, 0, z_value);
imag_integral = quadgk(imagPart_F, 0, z_value);

integral_F = real_integral + 1i * imag_integral;

disp(integral_F);

   0.4024 + 0.5536i

[simran]

But I don't want to fix n and z. I want the answer in terms of function of z, since 0 to z are the limits.

[Mrutyunjaya Hiremath]

% Example usage
n = 2;  % Replace with your desired value for n
z_target = 0.5 + 0.5i;  % Replace with your desired value for z
Fz_result = calculate_Fz(z_target, n);
disp(['F(', num2str(z_target), ') = ', num2str(Fz_result)]);

F(0.5+0.5i) = 0.54931

function Fz = calculate_Fz(z_target, n)
    % Define the real and imaginary parts of the integrand
    fun_real = @(t) real(1 ./ (1 - t.^n));
    fun_imag = @(t) imag(1 ./ (1 - t.^n));
    
    % Perform numerical integration using quadgk
    integral_real = quadgk(fun_real, 0, real(z_target));
    integral_imag = quadgk(fun_imag, 0, imag(z_target));
    
    % Combine the real and imaginary parts to get the complex integral result
    Fz = integral_real + 1i * integral_imag;
end

In this example, the function 'calculate_Fz' calculates F(z) for a given z and n by integrating from 0 to z. The real and imaginary parts of the integral are computed separately and then combined to get the final complex-valued result.

Simply run this code in MATLAB, and it will display F(z) for your chosen z and n.

[Torsten]

@Mrutyunjaya Hiremath

Your first code was correct.

n = 2;  % Replace with your desired value for n
z_target = 0.5 + 0.5*1i;  % Replace with your desired value for z
% Parametrize path from 0 to z
para = @(t) z_target*t;
dpara = @(t) z_target*ones(size(t));
f = @(z,n) 1./(1-z.^n);
fun_to_integrate = @(t,n) f(para(t),n).*dpara(t);
Fz_result = integral(@(t)fun_to_integrate(t,n),0,1)

Fz_result = 0.4024 + 0.5536i

syms z
f = 1/(1-z^n);
Fz_result = int(f,z,0,z_target);
double(Fz_result)

ans = 0.4024 + 0.5536i

[Walter Roberson]

@Mrutyunjaya Hiremath

But I don't want to fix n and z. I want the answer in terms of function of z, since 0 to z are the limits.

quadgk() is strictly a numeric integration routine. No numeric integration routine is going to be able to return a symbolic function.

Answer 3

[David Goodmanson]

Hi simran,

for any general function f(z) and a set of points [z1, z2, z3 ...] that describes a path in the complex plane (not necessarily a closed path) then numerically you have

function I = contourint(z,f)
% 
% complex contour integral by simple trapezoidal method.
% z is a path in the complex plane, f(z) is the integrand
% path does not have to be closed.
%
% I = contourint(z,f)
I = (1/2)*sum((f(1:end-1)+f(2:end)).*diff(z));

Of course you want the z points to be closely spaced enough to describe the path well, and also closely spaced enough to describe f(z) well, if f(z) varies quickly along the path. A typical case of that is when the path goes close by to a pole in f(z), and you need a denser set of z points in that section to pass by the pole. (No need for all the z points to be equally spaced along the path in terms of the distance between them). Don't be afraid to use lots of points, many thousands or more work well before you reach the point of diminishing returns.

This all makes the most sense when f(z) is analytic, but numerically there is not even a requirement for that.

Here are some simple examples where the path is closed and runs around the unit circle

n = 1e5;
th = ((0:n)/n)*2*pi;
z = exp(i*th);            % closed path around the unit circle

zin = (1/2)+i*(1/3);      % first order pole inside the path
f = 1./(z-zin);            
I1 = contourint(z,f)

zout = 2+3*i;             % first order pole outside the path
f = 1./(z-zout);            
I2 = contourint(z,f)

f = 1./(z-zin).^2;         % second order pole inside the path
I3 = contourint(z,f)

f = (z-zin).^3 + (z-zout).^2;  % no singularities inside the path
I4 = contourint(z,f)

% results
I1 = 0.0000 + 6.2832i             % = 2*pi*i*residue  ok 
% all these are zero
I2 = -2.2204e-16 - 1.1102e-16i   
I3 = -6.6613e-16 - 1.6931e-15i
I4 = -8.8818e-15 + 1.7764e-15i

% example, integral for bessel function J1(a)
a = 2;
f = exp((z-1./z)*a/2)./(z.^2);
J1a = (1/(2*pi*i))*contourint(z,f)
% check
J1a_exact = besselj(1,a)
J1a - J1a_exact

  J1a = 0.5767 - 0.0000i
  J1a_exact = 0.5767
  ans = -3.7947e-10 - 4.4174e-17i