Cauchy-Riemann Equations - Complex Analysis, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Complex derivatives

Having discussed some of the basic properties of functions, we ask now what it means for a function to have a complex derivative. Here we will see something quite new: this is very different from asking that its real and imaginary parts have partial derivatives with respect to x and y. We will not worry about the meaning of the derivative in terms of slope, but only ask that the usual difference quotient exists.

Denition : A function f (z) is complex differentiable at c if

exists. In this case, the limit is denoted by f ' (c). Making the change of variable z = c + h, f (z) is complex differentiable at c if and only if the limit

exists, in which case the limit is again f ' (c). A function is complex differentiable if it is complex differentiable at every point where it is dened. For such a function f (z), the derivative denes a new function which we write as 

For example, a constant function f (z) = C is everywhere complex differentiable and its derivative f ' (z) = 0. The function f (z) = z is also complex differentiable, since in this case

Thus (z)0 = 1. But many simple functions do not have complex derivatives. For example, consider f (z) = Re z = x. We show that the limit

does not exist for any c. Let c = a + bi, so that f (c) = a. First consider h = t a real number. Then f (c + t) = a + t and so

So if the limit exists, it must be 1. On the other hand, we could use h = it. In this case, f (c + it) = f (c) = a, and

Thus approaching c along horizontal and vertical directions has given two different answers, and so the limit cannot exist. Other simple functions which can be shown not to have complex derivatives are Im z; , and |z|. 
On the bright side, the usual rules for derivatives can be checked to hold:

1. If f (z) is complex differentiable, then so is cf (z), where c is a constant, and (cf (z))' = cf ' (z);

2. (Sum rule) If f (z) and g(z) are complex differentiable, then so is f (z)+ g(z), and (f (z) + g(z))' = f ' (z) + g' (z);

3. (Product rule) If f (z) and g(z) are complex differentiable, then so is f (z) . g(z) and (f (z) . g(z))' = f ' (z)g(z) + f (z)g' (z);

4. (Quotient rule) If f (z) and g(z) are complex differentiable, then so is f (z)/g(z), where dened (i.e. where g(z) ≠ 0), and

5. (Chain rule) If f (z) and g(z) are complex differentiable, then so is f (g(z)) where dened, and (f (g(z )))' = f '(g(z)) . g' (z).

6. (Inverse functions) If f (z) is complex differentiable and one-to-one, with nonzero derivative, then the inverse function f ^-1 (z) is also differentiable, and

(f ^-1 (z))' = 1/f ' (f ^-1(z)):

Thus for example we have the power rule (zn)0 = nzn 1, every polynomial P (z) = a_nzⁿ + a_n-1z^{n -1} + ... + a₀ is complex differentiable, with P '(z) = na_nz^n-1 + (n - 1)a_{n - 1}z^{n - 2} ... + a₁, and every rational function is also complex differentiable. It follows that a function which is not complex differentiable, such as Re z or  cannot be written as a complex polynomial or rational function.

The Cauchy-Riemann equations

We now turn systematically to the question of deciding when a complex function f (z) = u + iv is complex differentiable. If the complex derivative f '(z) is to exist, then we should be able to compute it by approaching z along either horizontal or vertical lines. Thus we must have

where t is a real number. In terms of u and v,

Taking the derivative along a vertical line gives

Equating real and imaginary parts, we see that: If a function f (z) = u + iv is complex differentiable, then its real and imaginary parts satisfy the Cauchy-Riemann equations:

Moreover, the complex derivative f '(z) is then given by

Examples: the function z² = (x² - y²) + 2xyi satis es the Cauchy- Riemann equations, since

Likewise, e^z = e^x cos y + ie^x sin y satis es the Cauchy-Riemann equations, since

Moreover, e^z is in fact complex differentiable, and its complex derivative is

The chain rule then implies that, for a complex number   One can dene cos z and sin z in terms of e^iz and e^-iz (see the homework). From the sum rule and the expressions for cos z and sin z in terms of eiz and e iz , it is easy to check that cos z and sin z are analytic and that the usual rules hold:

On the other hand, does not satisfy the Cauchy-Riemann equations, since

Likewise, f (z) = x² +iy² does not. Note that the Cauchy-Riemann equations are two equations for the partial derivatives of u and v, and both must be satised if the function f (z) is to have a complex derivative.
We have seen that a function with a complex derivative satises the Cauchy-Riemann equations. In fact, the converse is true:

Theorem: Let f (z) = u + iv be a complex function dened in a region (open subset) D of C, and suppose that u and v have continuous rst partial derivatives with respect to x and y. If u and v satisfy the Cauchy-Riemann equations, then f (z) has a complex derivative.

The proof of this theorem is not dicult, but involves a more careful understanding of the meaning of the partial derivatives and linear approximation in two variables.

Thus we see that the Cauchy-Riemann equations give a complete criterion for deciding if a function has a complex derivative. There is also a geometric interpretation of the Cauchy-Riemann equations. Recall that  Then u and v satisfy the Cauchy-Riemann equations if and only if

If this holds, then the level curves u = c₁ and v = c₂ are orthogonal where they intersect.

Instead of saying that a function f (z) has a complex derivative, or equivalently satises the Cauchy-Riemann equations, we shall call f (z) analytic or holomorphic. Here are some basic properties of analytic functions, which are easy consequences of the Cauchy-Riemann equations:

Theorem: Let f (z) = u + iv be an analytic function.

1. If f ' (z) is identically zero, then f (z) is a constant.

2. If either Re f (z) = u or Im f (z) = v is constant, then f (z) is constant.
In particular, a nonconstant analytic function cannot take only real or only pure imaginary values.

3. If jf (z)j is constant or arg f (z) is constant, then f (z) is constant.

For example, if f ' (z) = 0, then

Thus   By the Cauchy-Riemann equations,  as well. Hence f (z) is a constant. This proves (1). To see (2), assume for instance that u is constant.  Then   and, as above, the Cauchy-Riemann equations then imply that  Again, f (z) is constant. Part (3) can be proved along similar but more complicated lines.

Harmonic functions

Let f (z) = u + iv be an analytic function, and assume that u and v have partial derivatives of order 2 (in fact, this turns out to be automatic). Then, using the Cauchy-Riemann equations and the equality of mixed partials, we have:

In other words, u satis es:

The above equation is a very important second order partial differential equation, and solutions of it are called harmonic functions. Thus, the real part of an analytic function is harmonic. A similar argument shows that v is also harmonic, i.e. the imaginary part of an analytic function is harmonic. Essentially, all harmonic functions arise as the real parts of analytic functions.

Theorem: Let D be a simply connected region in C and let u(x; y) be a realvalued, harmonic function in D. Then there exists a real-valued function v(x; y) such that f (z) = u + iv is an analytic function.

We will discuss the meaning of the simply connected condition in the exercises in the next handout. The problem is that, if D is not simply connected, then it is possible that u can be completed to an analytic "function" f (z) = u + iv which is not single-valued, even if u is single valued. The basic example is Re log z = 1/2 ln(x² + y²). A calculation (left as homework) shows that this function is harmonic. But an analytic function whose real part is the same as that of log z must agree with log z up to an imaginary constant, and so cannot be single-valued.
The point to keep in mind is that we can generate lots of harmonic functions, in fact essentially all of them, by taking real or imaginary parts of analytic functions. Harmonic functions are very important in mathematical physics, and one reason for the importance of analytic functions is their connection to harmonic functions.