Cauchy-Riemann Equations: Definition & Examples

Cauchy-Riemann Equations: Definition & Examples
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  • 0:04 Cauchy-Riemann Equations
  • 0:36 Derivative of a…
  • 2:43 Using Cauchy-Riemann
  • 5:30 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

The derivative of a complex function may or may not exist. To find out, we use test equations. In this lesson, we will derive and use the Cauchy-Riemann equations and then apply these tests to several examples.

Cauchy-Reimann Equations

''To be or not to be?'' is a complex question. It's 10 a.m. and Jason is still asleep. Will he or will he not be late for his 8 a.m. class? This is a question.

''Will a complex function have or not have a complex derivative?'' sounds complicated, but there is an easy way to figure it out.

In this lesson, we'll answer this question and derive the conditions for the existence of the complex derivative. These conditions are called the Cauchy-Riemann equations. They are almost like the conditions for Jason to be at his next class on time.

Derivative of a Complex Function

We are going to talk about f(z) = u + iv, which is a complex function. We know this because the function f(z) has a real part (u) and an imaginary part (v). In general, both u and v are functions of x and y. We also know how to define the derivative for real functions. So, what do we mean by the derivative of a complex function?

Analogous to the definition of the derivative for real functions, we can write:

derivative_of_a_complex_function

Substituting for u + iv for f(x) and taking the limit along the real axis gives:

along_the_real_axis

Now, we group the real parts and the imaginary parts:

grouping_real_and_imaginary_parts

We identify partial derivatives with respect to x. A partial derivative is like an ordinary derivative except one letter is allowed to vary (x in this case) and all the other letters are held constant. Keeping y constant is the reason we are evaluating along the real axis. Using partial derivatives:

partial_derivatives_wrt_x

The letter u with a subscript x is another way to write the partial derivative of u with respect to x. Likewise, the partial derivative of v with respect to x is written as the letter v with a subscript x.

Repeat these steps for the limit along the imaginary axis:

along_the_imaginary_axis

Again, grouping the real and imaginary parts:

grouping_real_and_imaginary

Identifying partial derivatives with respect to y:

partial_derivatives_wrt_y

Back in the limit equations, do you see where the ''i'' in the denominator became ''-i'' in the numerator of the first term? In the second term, the ''i'' in the numerator and denominator cancelled.

We now have two results for the derivative of f(z):

f_prime_of_z_along_x

and

f_prime_of_z_along_y

These two derivative expressions can only be true if their real parts are equal and if their imaginary parts are equal. Equating the real parts:

the_first_CR_equations

Equating the imaginary parts:

the_second_CR_equations

These equations are called the Cauchy-Riemann equations. When these equations are true for a particular f(z), the complex derivative of f(z) exists.

Note the second equation is equivalent to and sometimes written as:

the_alternate_form_second_CR_equations

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