Download Complex Analysis: Functions of Complex Variables and Their Derivatives and more Study notes Complex analysis in PDF only on Docsity!
2 Complex Functions and the Cauchy-Riemann
Equations
2.1 Complex functions
In one-variable calculus, we study functions f (x) of a real variable x. Like- wise, in complex analysis, we study functions f (z) of a complex variable z ∈ C (or in some region of C). Here we expect that f (z) will in general take values in C as well. However, it will turn out that some functions are better than others. Basic examples of functions f (z) that we have already seen are: f (z) = c, where c is a constant (allowed to be complex), f (z) = z, f (z) = ¯z, f (z) = Re z, f (z) = Im z, f (z) = |z|, f (z) = ez^. The “func- tions” f (z) = arg z, f (z) =
z, and f (z) = log z are also quite interesting, but they are not well-defined (single-valued, in the terminology of complex analysis). What is a complex valued function of a complex variable? If z = x + iy, then a function f (z) is simply a function F (x, y) = u(x, y) + iv(x, y) of the two real variables x and y. As such, it is a function (mapping) from R^2 to R^2. Here are some examples:
- f (z) = z corresponds to F (x, y) = x + iy (u = x, v = y);
- f (z) = ¯z, with F (x, y) = x − iy (u = x, v = −y);
- f (z) = Re z, with F (x, y) = x (u = x, v = 0, taking values just along the real axis);
- f (z) = |z|, with F (x, y) =
x^2 + y^2 (u =
x^2 + y^2 , v = 0, taking values just along the real axis);
- f (z) = z^2 , with F (x, y) = (x^2 − y^2 ) + i(2xy) (u = x^2 − y^2 , v = 2xy);
- f (z) = ez^ , with F (x, y) = ex^ cos y + i(ex^ sin y) (u = ex^ cos y, v = ex^ sin y).
If f (z) = u + iv, then the function u(x, y) is called the real part of f and v(x, y) is called the imaginary part of f. Of course, it will not in general be possible to plot the graph of f (z), which will lie in C^2 , the set of ordered pairs of complex numbers, but it is the set {(z, w) ∈ C^2 : w = f (z)}. The graph can also be viewed as the subset of R^4 given by {(x, y, s, t) : s = u(x, y), t = v(x, y)}. In particular, it lies in a four-dimensional space. The usual operations on complex numbers extend to complex functions: given a complex function f (z) = u+iv, we can define functions Re f (z) = u,
Im f (z) = v, f (z) = u − iv, |f (z)| =
u^2 + v^2. Likewise, if g(z) is another complex function, we can define f (z)g(z) and f (z)/g(z) for those z for which g(z) 6 = 0. Some of the most interesting examples come by using the algebraic op- erations of C. For example, a polynomial is an expression of the form
P (z) = anzn^ + an− 1 zn−^1 + · · · + a 0 ,
where the ai are complex numbers, and it defines a function in the usual way. It is easy to see that the real and imaginary parts of a polynomial P (z) are polynomials in x and y. For example,
P (z) = (1 + i)z^2 − 3 iz = (x^2 − y^2 − 2 xy + 3y) + (x^2 − y^2 + 2xy − 3 x)i,
and the real and imaginary parts of P (z) are polynomials in x and y. But given two (real) polynomial functions u(x, y) and z(x, y), it is very rarely the case that there exists a complex polynomial P (z) such that P (z) = u + iv. For example, it is not hard to see that x cannot be of the form P (z), nor can ¯z. As we shall see later, no polynomial in x and y taking only real values for
every z (i.e. v = 0) can be of the form P (z). Of course, since x =
(z + ¯z)
and y =
2 i (z − ¯z), every polynomial F (x, y) in x and y is also a polynomial
in z and ¯z, i.e. F (x, y) = Q(z, z¯) =
i,j≥ 0
cij zi^ z¯j^ ,
where cij are complex coefficients. Finally, while on the subject of polynomials, let us mention the
Fundamental Theorem of Algebra (first proved by Gauss in 1799): If P (z) is a nonconstant polynomial, then P (z) has a complex root. In other words, there exists a complex number c such that P (c) = 0. From this, it is easy to deduce the following corollaries:
- If P (z) is a polynomial of degree n > 0, then P (z) can be factored into linear factors:
P (z) = a(z − c 1 ) · · · (z − cn),
for complex numbers a and c 1 ,... , cn.
- Every nonconstant polynomial p(x) with real coefficients can be fac- tored into (real) polynomials of degree one or two.
2.2 Limits and continuity
The absolute value measures the distance between two complex numbers. Thus, z 1 and z 2 are close when |z 1 − z 2 | is small. We can then define the limit of a complex function f (z) as follows: we write
zlim→c f^ (z) =^ L,
where c and L are understood to be complex numbers, if the distance from f (z) to L, |f (z) − L|, is small whenever |z − c| is small. More precisely, if we want |f (z) − L| to be less than some small specified positive real number , then there should exist a positive real number δ such that, if |z − c| < δ, then |f (z) − L| < . Note that, as with real functions, it does not matter if f (c) = L or even that f (z) be defined at c. It is easy to see that, if c = (c 1 , c 2 ), L = a + bi and f (z) = u + iv is written as a real and an part, then limz→c f (z) = L if and only if lim(x,y)→(c 1 ,c 2 ) u(x, y) = a and lim(x,y)→(c 1 ,c 2 ) v(x, y) = b. Thus the story for limits of functions of a complex variable is the same as the story for limits of real valued functions of the variables x, y. However, a real variable x can approach a real number c only from above or below (or from the left or right, depending on your point of view), whereas there are many ways for a complex variable to approach a complex number c. Sequences, limits of sequences, convergent series and power series can be defined similarly. As for functions of a real variable, a function f (z) is continuous at c if
lim z→c f (z) = f (c).
In other words: 1) the limit exists; 2) f (z) is defined at c; 3) its value at c is the limiting value. A function f (z) is continuous if it is continuous at all points where it is defined. It is easy to see that a function f (z) = u + iv is continuous if and only if its real and imaginary parts are continuous, and that the usual functions z, z,¯ Re z, Im z, |z|, ez^ are continuous. (We have to be careful, though, about functions such as arg z or log z which are not well-defined.) All polynomials P (z) are continuous, as are all two-variable polynomial functions in x and y. A rational function R(z) = P (z)/Q(z) with Q(z) not identically zero is continuous where it is defined, i.e. at the finitely many points where the denominator Q(z) is not zero. More generally, if f (z) and g(z) are continuous, then so are:
- cf (z), where c is a constant;
- f (z) + g(z);
- f (z) · g(z);
- f (z)/g(z), where defined (i.e. where g(z) 6 = 0).
- (g ◦ f )(z) = g(f (z)), the composition of g(z) and f (z), where defined.
2.3 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.
Definition A function f (z) is complex differentiable at c if
lim z→c
f (z) − f (c) z − c
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
lim h→ 0
f (c + h) − f (c) h
exists, in which case the limit is again f ′(c). A function is complex differen- tiable if it is complex differentiable at every point where it is defined. For such a function f (z), the derivative defines a new function which we write
as f ′(z) or d dz
f (z).
For example, a constant function f (z) = C is everywhere complex differ- entiable and its derivative f ′(z) = 0. The function f (z) = z is also complex differentiable, since in this case
f (z) − f (c) z − c
z − c z − c
Thus (z)′^ = 1. But many simple functions do not have complex derivatives. For example, consider f (z) = Re z = x. We show that the limit
lim h→ 0
f (c + h) − f (c) h
2.4 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
f ′(z) = lim t→ 0
f (z + t) − f (z) t = lim t→ 0
f (z + it) − f (z) it
where t is a real number. In terms of u and v,
lim t→ 0
f (z + t) − f (z) t = lim t→ 0
u(x + t, y) + iv(x + t, y) − u(x, y) − v(x, y) t
= lim t→ 0
u(x + t, y) − u(x, y) t
v(x + t, y) − v(x, y) t
∂u ∂x
Taking the derivative along a vertical line gives
lim t→ 0
f (z + it) − f (z) it = −i lim t→ 0
u(x, y + t) + iv(x, y + t) − u(x, y) − v(x, y) t
= −i lim t→ 0
u(x, y + t) − u(x, y) t
v(x, y + t) − v(x, y) t
= −i ∂u ∂y
∂v ∂y
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:
∂u ∂x
∂v ∂y
∂v ∂x
∂u ∂y
Moreover, the complex derivative f ′(z) is then given by
f ′(z) =
∂u ∂x
∂v ∂x
∂v ∂y − i
∂u ∂y
Examples: the function z^2 = (x^2 − y^2 ) + 2xyi satisfies the Cauchy- Riemann equations, since
∂ ∂x
(x^2 − y^2 ) = 2x =
∂y
(2xy) and
∂x
(2xy) = 2y = −
∂y
(x^2 − y^2 ).
Likewise, ez^ = ex^ cos y + iex^ sin y satisfies the Cauchy-Riemann equations, since ∂ ∂x (ex^ cos y) = ex^ cos y =
∂y (ex^ sin y) and
∂x (ex^ sin y) = ex^ sin y = −
∂y (ex^ cos y).
Moreover, ez^ is in fact complex differentiable, and its complex derivative is
d dz
ez^ =
∂x
(ex^ cos y) +
∂x
(ex^ sin y) = ex^ cos y + ex^ sin y = ez^.
The chain rule then implies that, for a complex number α, d dz
eαz^ = αeαz^.
One can define cos z and sin z in terms of eiz^ 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: d dz cos z = − sin z;
d dz sin z = cos z.
On the other hand, ¯z does not satisfy the Cauchy-Riemann equations, since
∂ ∂x (x) = 1 6 =
∂y (−y).
Likewise, f (z) = x^2 +iy^2 does not. Note that the Cauchy-Riemann equations are two equations for the partial derivatives of u and v, and both must be satisfied if the function f (z) is to have a complex derivative. We have seen that a function with a complex derivative satisfies the Cauchy-Riemann equations. In fact, the converse is true:
Theorem: Let f (z) = u + iv be a complex function defined in a region (open subset) D of C, and suppose that u and v have continuous first 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 difficult, but involves a more careful understanding of the meaning of the partial derivatives and linear approxi- mation in two variables. Thus we see that the Cauchy-Riemann equations give a complete cri- terion for deciding if a function has a complex derivative. There is also a geometric interpretation of the Cauchy-Riemann equations. Recall that
∇u =
∂u ∂x
∂u ∂y
and that ∇v =
∂v ∂x
∂v ∂y
. Then u and v satisfy the
Cauchy-Riemann equations if and only if
∇v =
∂v ∂x
∂v ∂y
∂u ∂y
∂u ∂x
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 har- monic. 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 real- valued, 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 con- nected, 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 = 12 ln(x^2 + y^2 ). 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.
2.6 Homework
- Write the function f (z) in the form u + iv:
(a) z + iz^2 ; (b) 1/z; (c) ¯z/z.
- If f (z) = ez^ , describe the images under f (z) of horizontal and vertical lines, i.e. what are the sets f (a + it) and f (t + ib), where a and b are constants and t runs through all real numbers?
- Is the function ¯z/z continuous at 0? Why or why not? Is the function z/z¯ analytic where it is defined? Why or why not?
- Compute the derivatives of the following analytic functions, and be prepared to justify your answers:
(a) iz + 3 z^2 − (2 + i)z + (4 − 3 i) ; (b) ez 2 ; (c)
ez^ + e−z^
- Let f (z) be a complex function. Is it possible for both f (z) and f (z) to be analytic?
- Let f (z) = u + iv be analytic. Recall that the Jacobian is the function given by the following determinant:
∂(u, v) ∂(x, y)
∂u/∂x ∂u/∂y ∂v/∂x ∂v/∂y
Using the Cauchy-Riemann equations, show that this is the same as |f ′(z)|^2.
- Define the complex sine and cosine functions as follows:
cos z =
(eiz^ + e−iz^ ); sin z =
2 i (eiz^ − e−iz^ ).
Note that, if t is real, then this definition of cos t and sin t agree with the usual ones, and that (for those who remember hyperbolic func- tions) cos z = cosh iz and sin z = −i sinh iz. Verify that cos z and sin z are analytic and that (cos z)′^ = − sin z and that (sin z)′^ = cos z. Write cos z as u + iv, where u and v are real-valued functions of x and y, and similarly for sin z.
- Verify that Re 1/z, Im 1/z, and Re log z = 12 ln(x^2 + y^2 ) are harmonic.
- Which of the following are harmonic?
(a) x^3 − y^3 ; (b) x^3 y − xy^3 ; (c) x^2 − 2 xy.
- If f (z) = u + iv is a complex function such that u and v are both harmonic, is f (z) necessarily analytic?
