Calculus:Complex analysis
Complex analysis is the study of functions of complex variables. Complex analysis is a widely used and powerful tool in certain areas of electrical engineering, and others.
Before we begin, you may want to review Complex numbers
Complex Numbers
Complex Numbers
Complex Functions
A function of a complex variable is a function that can take on complex values, as well as strictly real ones. For example, suppose f(z) = z2. This function sets up a correspondence between the complex number z and its square, z2, just like a function of a real variable, but with complex numbers.
Note that, for f(z) = z2, f(z) will be strictly real if z is strictly real.
Generally we can write a function f(z) in the form f(z) = f(x+iy) = a(x,y) + ib(x,y), where a and b are real-valued functions.
Limits and continuity
As with real-valued functions, we have concepts of limits and continuity with complex-valued functions also – our usual delta-epsilon limit definition:
- The limit of f(z) as z approaches w is L if for each ε > 0, there is a δ > 0 such that | f(z)-L |<ε for all z such that 0 < | z - w | < δ.
Note that ε and δ are real values. This is implicit in the use of inequalities: only real values are "greater than zero".
One difference between this definition of limit and the definition for real-valued functions is the meaning of the absolute value. Here we mean the complex absolute value instead of the real-valued one. Another difference is that of how z approaches w. For real-valued functions, we would only be concerned about z approaching w from the left, or from the right. In a complex setting, z can approach w from any direction in the two-dimensional complex plane: along any line passing through w, along a spiral centered at w, etc.
For example, let f(z) = z2. Suppose we want to show that the limit of f(z) as z approaches i is -1. We can write z as i+γ where we think of
γ being a small complex quantity. Note then that z-i = γ. Then, with L in our definition being -1, and w being i, we have
- | f(z) - L | = | z2 + 1 | = | (i + γ)2 + 1 | = | 2i γ + γ2 |
By the triangle inequality, this last expression is less than
- 2 | γ | + | γ |2
In order for this to be less than ε, we can require that
- | γ | < 1/2 min( ε/2, √ ε)
Thus, for any ε > 0, if δ = 1/2 min( ε/2, √ ε), and | z - i |<δ, then | f(z) - ( - 1) | < ε.
Hence, the limit of f(z)=z2 as z approaches i is -1.
Differentiation and Holomorphic Functions
Since we have limits defined, we can go ahead to define the derivative of a complex function, in the usual way:
-
provided that the limit is the same no matter how Δz approaches zero (since we are working now in the complex plane, we have more freedom!).
If such a limit exists for some value z, or some set of values - a region, we call the function holomorphic at that point or region. Continuity and being single-valued are necessary for being analytic; however, continuity and being single-valued are not sufficient for being analytic.
Many elementary functions of complex values have the same derivatives as those for real functions: for example D z2 = 2z.
Problem set
Given the above, answer the following questions (Answers follow to even-numbered questions).
- Find the derivative of z3 from the limit definition.
- Write ez in the form a(x, y)+b(x, y)i
Answers
1. 
2. 
Cauchy-Riemann Equations
We might wonder which sorts of complex functions are in fact differentiable. It would appear that the criterion for holomorphicity is much stricter than that of differentiability for real functions, and this is indeed the case. Suppose we have a complex function
-
 ,
where u and v are real functions. Assume furthermore that u and v are differentiable functions in the real sense. Then we can let Δz in the definition of differentiability approach 0 by varying only x or only y. Therefore f can only be differentiable in the complex sense if
In fact, if u and v are differentiable in the real sense and satisfy these two equations, then f is holomorphic. These two equations are known as the Cauchy-Riemann equations.
Integration
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