Introduction to the Derivative of Complex Functions

A function containing a real and an imaginary part is known as a complex function. Since calculus involves the study of rate of change, it also involves the study of rate of change of complex functions. It can be calculated by using different techniques of differentiation. Let’s understand complex function differentiation and its method of calculation. Before this, let’s discuss what complex functions are.

What are complex functions?

The understanding of the complex functions is important to understand the complex differentiation. A complex function can be defined as a relation between two complex numbers. In other words, it is a function that takes a complex number as an input and produces a complex number as an output. 

In mathematical terms, a complex number contains a real and an imaginary part which is expressed by ‘i’. It is written as;

$f(z) = u(x,y)+v(x,y)i$

Where,

• $z=x+iy{2}nbsp;
• $u(x,y)$ is the real part of the function f(z).
• $v(x,y)$ is the imaginary part of the function f(z).

What is the Derivative of a Complex Function?

The derivative of a complex function is the rate of change with respect to the real and imaginary parts involved. And the process of finding derivative of a complex function is known as complex function differentiation. In this process, we use the method of partial derivative to differentiate a complex function. 

A complex function is only differentiable if its graph reaches the same point from different paths. In other words, the complex function must be continuous to be differentiable. If it is not a continuous function, then it will not be differentiable. To understand the differentiation of complex functions, let’s discuss the complex differentiation formula.

Complex Differentiation Formula

Since we use the delta method to differentiate a function in calculus. This method can also be transformed into a formula to differentiate complex functions. For a complex function f(z), the derivative by definition formula can be written as;

$f’(z)=\lim_{h \to 0}\frac{f(z+h)-f(z)}{h}$

Where, h is the small complex number. This formula can be simplified by using the Cauchy-Riemann equation which states that the real and imaginary part of the complex function must satisfy the following conditions.

• $u_x=v_y$
• $u_y=-v_x$

Using these conditions, the derivative formula of a complex function can be written by using the partial derivative as;

$f'(z) = u_x + iv_x = v_y - iu_y$

You can also use our partial derivative calculator to differentiate a complex function with easy steps.

How to differentiate a complex function?

As we discussed above, a complex function can be differentiated by using partial derivatives and the Cauchy-Riemann conditions in a combined form. This process looks a little bit tricky. Therefore, here we provide you a step-by-step method of finding derivatives of complex functions. To do this, follow the given steps;

1. First, identify the given function. If the function contains a real and an imaginary part, continue to the second step.
2. Calculate the partial derivative of both real and imaginary parts with respect to x and y.
3. Now use the Cauchy-Riemann equation to calculate the derivative of the given complex function. 
4. Simplify if needed.

Let’s understand how to differentiate a complex function in the following example. 

Complex Function Differentiation Example

Consider that we want to calculate the derivative of f(z)=z^2 which can be written as;

$z^2=(x^2-y^2)+i(2xy)$

In this equation, the real part is,

$u=x^2-y^2$

And the imaginary part is,

$v=2xy$

Calculating the derivative of both real and imaginary parts with respect to x and y.

$u_x=2x$

$u_y=-2y$

Similarly, 

$v_x=2y$

$v_y=2x$

Now to calculate f’(z), we will use the following formula, 

$f'(z) = u_x + iv_x = v_y - iu_y$

Substituting the values of all derivatives, we get

$f'(z) = 2x + i2y = 2x + i2y$

We have, 

$f'(z) = 2(x+iy)$

Rules of Differentiation For Complex Functions

Just like real functions, complex functions also have rules of differentiation. These rules make it easier to find the derivative of a complex function. Here are some of the rules of differentiation for complex functions:

1. Power Rule: If f(z) = z^n, where n is a constant, then f'(z) = nz^(n-1).
2. Sum Rule: If f(z) = g(z) + h(z), then f'(z) = g'(z) + h'(z).
3. Product Rule: If f(z) = g(z)h(z), then f'(z) = g(z)h'(z) + g'(z)h(z).
4. Quotient Rule: If f(z) = g(z)/h(z), then f'(z) = [g'(z)h(z) - g(z)h'(z)]/h(z)^2.
5. Chain Rule: If f(z) = g(h(z)), then f'(z) = g'(h(z))h'(z).

Since the complex differentiation also uses the same derivative rules, you can use our chain rule calculator to find the solution of a complex function derivative.

Conclusion

In mathematics, the derivative of complex function is the rate of change of the real and imaginary parts. It is calculated by using ordinary derivative laws such as chain rule, product rule, power rule or quotient rule. For this, the Cauchy-Riemann equation is used as a formula. This derivative is used to analyse the complex problems in calculus.