MATLAB Digest - March 2004

Calculating Complex Integrals Using MATLAB

Although anyone who has taken a calculus course is familiar with integrals, many people are not familiar with the concept of complex integrals. This article explains one part of the concept of complex integrals, and shows how the MATLAB integration function quad, although not its intended use, can be used to calculate some of these. We also consider the int function provided by the Symbolic Math Toolbox and its ability to evaluate integrals of complex functions.

In addition, the class of non-analytic functions, which the quad and int functions are not designed to integrate, is considered. For this class of functions, the method of parameterization that can evaluate their integrals is introduced.

Real Integrals

The int and quad functions are two MATLAB integration functions. The quad function numerically calculates the definite integral of a function of one variable. The int function symbolically calculates the definite and indefinite integrals of a function over one of its variables.

Example 1

We can numerically calculate the definite integral

using

r = quad('3*sin(x/2)',0,1)
   
  r =

0.7345

Example 2

We can also symbolically calculate the indefinite integral

with the command

 r = int('a*sin(b*x)','x','x1','x2')

   
  r =

  -a*(cos(b*x2)-cos(b*x1))/b

Complex Integrals

If we consider z to be a complex variable (a variable that can have complex values), then the function f(z) is considered a complex function. The integral of a complex function is referred to as a complex integral, which is shown using the following notation

where C is the path, on the complex plane, over which the integral is to be calculated. When this path is a closed curve, this notation is given as

For a real integral, as in Example 1, the real variable x can have any value on the real axis between the two limits of integration. Therefore, we can say this integral is calculated over the straight line path from 0 to 1 on the real axis.

Figure 1

By definition, a complex variable z can be any value on the complex plane. Thus, when we discuss integrals of complex functions, we must define the path C on the complex plane over which this integral is to be performed. This complicates the process of evaluating complex integrals.

Analytic Functions

Consider the use of the quad function with complex values for the limits of integration. This computes the integral over a straight line path between the two specified endpoints.

Example 3

Calculate the complex integral

where C₁ is the straight line path from -1 + i to 2 - i.

Figure 2

Solution
The quad function is capable of performing this operation.

F1 = quad('sin(z)', -1+i, 2-i) %Calculate integral

F1 = 1.4759 - 0.0797i

We can compute the integral over C₃ in Figure 2 by summing two straight line integrals.

f = 'sin(z)'; %Define function

  F2 = quad(f, -1+i, 2+i) + quad(f, 2+i, 2-i) %Calculate integral

  

  F2 =

  1.4759 - 0.0797i

We find that F2 is the same as F1 for at least 4 decimal places. It can be shown that the integral of this function between these two endpoints will theoretically be the same regardless of whether the path is a straight line, a curved line, or a series of connected lines. Therefore, quad can be used to evaluate this integral over the path C₂ in Figure 2 by only specifying its endpoints, as was done for F1.

This is a consequence of the fact that this function is analytic¹ on the entire complex plane. A function f(z) is analytic over a domain R if f'(z) exists at every point within R.

Assume a function f(z) is analytic over a domain R, which is simply connected¹ (has no holes). Also, assume C₁ and C₂ are any two arbitrary paths between the point Z₁ and Z₂ within R, then

This is referred to as path independence¹.

Example 4

Calculate the integral of the complex function

over the paths C₁, C₂, and C₃ in Figure 3.

Figure 3

Solution
The function f(z) is differentiable at all points on the complex plane, except at the singular point
1 + 2i. Therefore, f(z) is analytic within any domain that does not contain 1 + 2i.

f = '(z + 1)./(z - 1 - 2i)'; %Define function

  F1 = quad(f, 0, 2) + quad(f, 2, 2+4i) %Compute integral



  
F1 =

-4.2832 +10.2832i

We find that the integrals over C₁ and C₂ are equal for at least 4 decimal places.

F2 = quad(f, 0, 1.5+i) + quad(f, 1.5+i, 2+4i) %Compute integral



 
   
F2 =

-4.2832 +10.2832i

This is due to the fact that there is a simply connected domain that contains both 0 and 2 + 4i, but does not contain 1 + 2i.

However, we find the integral over C₃ produces a different result.

F3 = quad(f, 0, 1+4i) + quad(f, 1+4i, 2+4i) %Compute integral
 
   

  F3 =

8.2832 - 2.2832i

This is due to the fact that there is no simply connected domain that contains C₁ and C₂, that does not contain 1 + 2i.

The result of this is that the integral of f(z), between 0 and 2 + 4i, is purely dependent on whether the path traverses clockwise or counter-clockwise around 1 + 2i.

Note that the following command returns a warning and a result with NaN values.

F = quad(f, 0, 2+4i) %Compute integral


   
  Warning: Divide by zero.

F =

NaN +    NaNi

This is due to the fact that 1 + 2i lies on the straight line path between 0 and 2 + 4i, and f(1 + 2i) does not exist.

However, the int function is able to recognize this fact, and, for this example, returns the value of the integral over a counter-clockwise path around the singular point.

syms z %Define symbolic variable

  f = (z+1)/(z-1-2*i); %Define symbolic function

  F = int(f, 0, 2+4i); %Compute integral

  disp(double(F)) %Display the double precision value of the result



 -4.2832 +10.2832i

Nonanalytic Functions

There are many complex functions that are not analytic anywhere on the complex plane. Therefore, in general, the integrals of these functions, between any two specified endpoints, will be different depending on the path taken between them. For these situations, we need to be able to perform the integral over a specific path.

For example, consider the function

where denotes the conjugate of z. Since f(z) involves , it is neither differentiable nor analytic anywhere on the complex plane. Therefore, we cannot assume the integral of f(z) over any path will be the same as that over the straight line path, as computed by the quad function.

Although we can use quad to approximate a curved path using a series of connected lines, this method is subject to a high degree of error. For these situations, we must utilize the parametric representation of this path.

Parametric Representations of the Path

Consider a curve C on the complex plane. We may be able to parameterize this curve by introducing a real parameter t such that every single point on C corresponds to a unique value of
a < t < b, where the values t = a and t = b correspond to the curve’s endpoints.

For example, a circle on the complex plane can be represented as C: z(x,y) = x + iy where
x² + y² = 1. This circle can be parameterized by the substitutions x(t) = cos(t) and y(t) = sin(t), where t is a real parameter. Therefore, a path on this circle is defined by C: z(t) = x(t) + iy(t), where t goes from t₀ to t₁. If t₀ < t₁, then the path will be counter-clockwise along the circle.

Figure 4

Example 5

Find the parametric representation of the path C: z(x,y) = x + iy, where y² - x + 1 = 0, as shown in Figure 4.

Figure 4

Solution
We can introduce the substitution y = r where -2 < r < 2. Therefore, from the equation y² - x + 1 = 0, we find that x = r² - 1. This results in the parametric representation of the path C: z(t) = r² + 1 + ir, where -2 < r < 2.

It is sometimes desirable to normalize the path so that z(0) = z₀ and z(1) = z₁. This can be achieved by generating C: z(t) where

, and 0 < t < 1. In this example, , therefore

C: z(t) = 16t² + (-16 + 4i)t + (5 - 2i)

We can generate the normalized parametric representation of this path using

syms t y %Define symbolic variable

  z0 = 5-2i; z1 = 5+2i; %Define endpoints

  f = y^2 + 1; %Define path

  y = imag(z0 + t*(z1 - z0)); %Generate y(t)

  x = subs(f, 'y', y); %Generate x(t)

  z = simple(x + i*y) %Generate z(t)

   

  z =

  5-16*t+16*t^2-2*i+4*i*t

The PathRep function (available on MATLAB Central) parameterizes a path specified as a function of x or y.

PATHREP generates the parametric representation of a path on the complex
    plane using the real parameter t over the range 0<=t<=1.


 
  z = PathRep(zi, zf, f) where


zi is the initial point of the path.

zf is the final point of the path.

f is the path given either as y(x) or x(y).  By default,

the path is assumed to be a straight line.


z is the parametric representation given as a function of t.

For this example, we can generate the parametric representation using

z = PathRep(5-2i, 5+2i, 'y^2 + 1') %Determine parametric representation

Integral Evaluation via Parametric Representations

Using the parametric representation z(t) of the path C, we can determine the integral of a function
f[z(t)] over this path via

where denotes the derivative of z(t) with respect to t.

Example 6

Evaluate

, where

and C is the path shown in Figure 6. Note that z - i

is a singular point of f(z), but is not included on the path.

Solution

We can use symbolic variables to determine the integrand.

syms t real %Define parameter

  z = PathRep(5-2i, 5+2i, 'y^2 + 1'); %Determine parameterized path

  f = conj(z)/(z-i); %Define function

  Integrand = f*diff(z,t); %Determine integrand

Then, either quad or int can be used to evaluate the integral. The quad function requires that the input function be given in vectorized form.

F = vectorize(char(Integrand)); %Create vectorized function

  Fq = quad(F, 0, 1) %Numerically evaluate integral


   
  Fq =


   
  1.4023 - 2.4213i


   
  Fi = int(Integrand, 't', 0, 1); %Symbolically evaluate integral

  Fi = double(Fi) %Convert to double precision value


   
  Fi =


   
  1.4023 - 2.4213i

Therefore, as expected, we find these two functions both return the same results, accurate to at least 4 decimal places.

Conclusion

The quad and int functions are intended to be used to compute real integrals. However, we can use these functions to directly compute some complex integrals over straight line paths. The integrals of analytic functions are independent of the path, therefore, the straight line integral computed by the quad function is sufficient to represent the integral over any path between the same endpoints.

For non-analytic functions, we can parameterize the path, and use Equation (1) to transform the complex integral into a real integral. The PathRep function can be used to determine the parametric representation of the path. Then, the quad or int functions can be used to compute this real integral.

Reference

¹ Esfandiari, Ramin, Applied Mathematics for Engineers, 3rd Ed., Atlantis Publishing, 2003.

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