RA Complex numbers

Section 11.1 Complex numbers

Note: half a lecture

Subsection 11.1.1 The complex plane

In this chapter we consider approximation of functions, or in other words functions as limits of sequences and series. We will extend some results we already saw to a somewhat more general setting, and we will look at some completely new results. In particular, we consider complex-valued functions. We gave complex numbers as examples before, but let us start from scratch and properly define the complex number field.

A complex number is just a pair \((x,y) \in \R^2\) on which we define multiplication (see below). We call the set the complex numbers and denote it by \(\C\text{.}\) We identify \(x \in \R\) with \((x,0) \in \C\text{.}\) The \(x\)-axis is then called the real axis and the \(y\)-axis is called the imaginary axis. As \(\C\) is just the plane, we also call the set \(\C\) the complex plane.

Define:

\begin{equation*} (x,y) + (s,t) \coloneqq (x+s,y+t) , \qquad (x,y) (s,t) \coloneqq (xs-yt,xt+ys) . \end{equation*}

Under the identification above, we have \(0 = (0,0)\) and \(1 = (1,0)\text{.}\) These two operations make the plane into a field (exercise). We write a complex number \((x,y)\) as \(x+iy\text{,}\) where we define

Note that engineers use \(j\) instead of \(i\text{.}\)

\begin{equation*} i \coloneqq (0,1) . \end{equation*}

Notice that \(i^2 = (0,1)(0,1) = (0-1,0+0) = -1\text{.}\) That is, \(i\) is a solution to the polynomial equation

\begin{equation*} z^2+1=0 . \end{equation*}

From now on, we will not use the notation \((x,y)\) and use only \(x+iy\text{.}\) See Figure 11.1.

Figure 11.1. The points \(1\text{,}\) \(i\text{,}\) \(x\text{,}\) \(iy\text{,}\) and \(x+iy\) in the complex plane.

We generally use \(x,y,r,s,t\) for real values and \(z,w,\xi,\zeta\) for complex values, although that is not a hard and fast rule. In particular, \(z\) is often used as a third real variable in \(\R^3\text{.}\)

Definition 11.1.1.

Suppose \(z= x+iy\text{.}\) We call \(x\) the real part of \(z\text{,}\) and we call \(y\) the imaginary part of \(z\text{.}\) We write

\begin{equation*} \Re\, z \coloneqq x , \qquad \Im\, z \coloneqq y . \end{equation*}

Define complex conjugate as

\begin{equation*} \bar{z} \coloneqq x-iy , \end{equation*}

and define modulus as

\begin{equation*} \sabs{z} \coloneqq \sqrt{x^2+y^2} . \end{equation*}

Modulus is the complex analogue of the absolute value and has similar properties. For example, \(\sabs{zw} = \sabs{z} \, \sabs{w}\) (exercise). The complex conjugate is a reflection of the plane across the real axis. The real numbers are precisely those numbers for which the imaginary part \(y=0\text{.}\) In particular, they are precisely those numbers which satisfy the equation

\begin{equation*} z = \bar{z} . \end{equation*}

As \(\C\) is really \(\R^2\text{,}\) we let the metric on \(\C\) be the standard euclidean metric on \(\R^2\text{.}\) In particular,

\begin{equation*} \sabs{z} = d(z,0) , \qquad \text{and also} \qquad \sabs{z-w} = d(z,w) . \end{equation*}

So the topology on \(\C\) is the same exact topology as the standard topology on \(\R^2\) with the euclidean metric, and \(\sabs{z}\) is equal to the euclidean norm on \(\R^2\text{.}\) Importantly, since \(\R^2\) is a complete metric space, then so is \(\C\text{.}\) As \(\sabs{z}\) is the euclidean norm on \(\R^2\text{,}\) we have the triangle inequality of both flavors:

\begin{equation*} \sabs{z+w} \leq \sabs{z}+\sabs{w} \qquad \text{and} \qquad \big\lvert \sabs{z}-\sabs{w} \big\rvert \leq \sabs{z-w} . \end{equation*}

The complex conjugate and the modulus are even more intimately related:

\begin{equation*} \sabs{z}^2 = x^2+y^2 = (x+iy)(x-iy) = z \bar{z} . \end{equation*}

Remark 11.1.2.

There is no natural ordering on the complex numbers. In particular, no ordering that makes the complex numbers into an ordered field. Ordering is one of the things we lose when we go from real to complex numbers.

Subsection 11.1.2 Complex numbers and limits

Algebraic operations with complex numbers are continuous because convergence in \(\R^2\) is the same as convergence for each component, and we already know that the real algebraic operations are continuous. For example, write \(z_n = x_n + i\,y_n\) and \(w_n = s_n + i\,t_n\text{,}\) and suppose that \(\lim_{n\to\infty} z_n = z = x+i\,y\) and \(\lim_{n\to\infty} w_n = w = s+i\,t\text{.}\) Let us show

\begin{equation*} \lim_{n\to\infty} z_n w_n = zw . \end{equation*}

First,

\begin{equation*} z_n w_n = (x_ns_n-y_nt_n) + i(x_nt_n+y_ns_n) . \end{equation*}

The topology on \(\C\) is the same as on \(\R^2\text{,}\) and so \(x_n \to x\text{,}\) \(y_n \to y\text{,}\) \(s_n \to s\text{,}\) and \(t_n \to t\text{.}\) Hence,

\begin{equation*} \lim_{n\to\infty} (x_ns_n-y_nt_n) = xs-yt \qquad \text{and} \qquad \lim_{n\to\infty} (x_nt_n+y_ns_n) = xt+ys . \end{equation*}

As \((xs-yt)+i(xt+ys) = zw\text{,}\)

\begin{equation*} \lim_{n\to\infty} z_n w_n = zw . \end{equation*}

Similarly the modulus and the complex conjugate are continuous functions. We leave the remainder of the proof of the following proposition as an exercise.

Proposition 11.1.3.

Suppose \(\{ z_n \}_{n=1}^\infty\text{,}\) \(\{ w_n \}_{n=1}^\infty\) are sequences of complex numbers converging to \(z\) and \(w\) respectively. Then

\(\displaystyle \lim_{n\to \infty} z_n + w_n = z + w\text{.}\)
\(\displaystyle \lim_{n\to \infty} z_n w_n = z w\text{.}\)
Assuming \(w_n \not= 0\) for all \(n\) and \(w\not= 0\text{,}\) \(\displaystyle \lim_{n\to \infty} \frac{z_n}{w_n} = \frac{z}{w}\text{.}\)
\(\displaystyle \lim_{n\to \infty} \sabs{z_n} = \sabs{z}\text{.}\)
\(\displaystyle \lim_{n\to \infty} \bar{z}_n = \bar{z}\text{.}\)

As we have seen above, convergence in \(\C\) is the same as convergence in \(\R^2\text{.}\) In particular, a sequence in \(\C\) converges if and only if the real and imaginary parts converge. Therefore, feel free to apply everything you have learned about convergence in \(\R^2\text{,}\) as well as applying results about real numbers to the real and imaginary parts.

We also need convergence of complex series. Let \(\{ z_n \}_{n=1}^\infty\) be a sequence of complex numbers. The series

\begin{equation*} \sum_{n=1}^\infty z_n \end{equation*}

converges if the limit of partial sums converges, that is, if

\begin{equation*} \lim_{k\to\infty} \sum_{n=1}^k z_n \qquad \text{exists.} \end{equation*}

A series converges absolutely if \(\sum_{n=1}^\infty \sabs{z_n}\) converges.

We say a series is Cauchy if the sequence of partial sums is Cauchy. The following two propositions have essentially the same proofs as for real series and we leave them as exercises.

Proposition 11.1.4.

The complex series \(\sum_{n=1}^\infty z_n\) is Cauchy if for every \(\epsilon > 0\text{,}\) there exists an \(M \in \N\) such that for every \(n \geq M\) and every \(k > n\text{,}\) we have

\begin{equation*} \abs{ \sum_{j={n+1}}^k z_j } < \epsilon . \end{equation*}

Proposition 11.1.5.

If a complex series \(\sum_{n=1}^\infty z_n\) converges absolutely, then it converges.

The series \(\sum_{n=1}^\infty \sabs{z_n}\) is a real series. All the convergence tests (ratio test, root test, etc.) that talk about absolute convergence work with the numbers \(\sabs{z_n}\text{,}\) that is, they are really talking about convergence of series of nonnegative real numbers. You can directly apply these tests them without needing to reprove anything for complex series.

Subsection 11.1.3 Complex-valued functions

When we deal with complex-valued functions \(f \colon X \to \C\text{,}\) what we often do is to write \(f = u+i\,v\) for real-valued functions \(u \colon X \to \R\) and \(v \colon X \to \R\text{.}\)

Suppose we wish to integrate \(f \colon [a,b] \to \C\text{.}\) We write \(f = u+i\,v\) for real-valued \(u\) and \(v\text{.}\) We say that \(f\) is Riemann integrable if \(u\) and \(v\) are Riemann integrable, and in this case we define

\begin{equation*} \int_a^b f \coloneqq \int_a^b u + i \int_a^b v . \end{equation*}

We make the same definition for every other type of integral (improper, multivariable, etc.).

Similarly when we differentiate, write \(f \colon [a,b] \to \C\) as \(f = u+i\,v\text{.}\) Thinking of \(\C\) as \(\R^2\text{,}\) we say that \(f\) is differentiable if \(u\) and \(v\) are differentiable. For a function valued in \(\R^2\text{,}\) the derivative is represented by a vector in \(\R^2\text{.}\) Now a vector in \(\R^2\) is a complex number. In other words, we write the derivative as

\begin{equation*} f'(t) \coloneqq u'(t) + i \, v'(t) . \end{equation*}

The linear operator representing the derivative is the multiplication by the complex number \(f'(t)\text{,}\) so nothing is lost in this identification.

Exercises 11.1.4 Exercises

11.1.1.

Check that \(\C\) is a field.

11.1.2.

Prove that for \(z,w \in \C\text{,}\) we have \(\sabs{zw} = \sabs{z} \, \sabs{w}\text{.}\)

11.1.3.

Finish the proof of Proposition 11.1.3.

11.1.4.

Prove Proposition 11.1.4.

11.1.5.

Prove Proposition 11.1.5.

11.1.6.

Given \(x +iy\) define the matrix \(\left[ \begin{smallmatrix} x & -y \\ y & x \end{smallmatrix} \right]\text{.}\) Prove:

The action of this matrix on a vector \((s,t)\) is the same as the action of multiplying \((x+iy)(s+it)\text{.}\)
Multiplying two such matrices is the same multiplying the underlying complex numbers and then finding the corresponding matrix for the product. In other words, the field \(\C\) can be identified with a subset of the 2-by-2 matrices.
The matrix \(\left[ \begin{smallmatrix} x & -y \\ y & x \end{smallmatrix} \right]\) has eigenvalues \(x+iy\) and \(x-iy\text{.}\) Recall that \(\lambda\) is an eigenvalue of a matrix \(A\) if \(A-\lambda I\) (a complex matrix in our case) is not invertible, that is, if it has linearly dependent rows: one row is a (complex) multiple of the other.

11.1.7.

Prove the Bolzano–Weierstrass theorem for complex sequences. Suppose \(\{ z_n \}_{n=1}^\infty\) is a bounded sequence of complex numbers, that is, there exists an \(M\) such that \(\sabs{z_n} \leq M\) for all \(n\text{.}\) Prove that there exists a subsequence \(\{ z_{n_k} \}_{k=1}^\infty\) that converges to some \(z \in \C\text{.}\)

11.1.8.

Prove that there is no simple mean value theorem for complex-valued functions: Find a differentiable function \(f \colon [0,1] \to \C\) such that \(f(0) = f(1) = 0\text{,}\) but \(f'(t) \not= 0\) for all \(t \in [0,1]\text{.}\)
However, there is a weaker form of the mean value theorem as there is for vector-valued functions. Prove: If \(f \colon [a,b] \to \C\) is continuous and differentiable in \((a,b)\text{,}\) and for some \(M\text{,}\) \(\sabs{f'(x)} \leq M\) for all \(x \in (a,b)\text{,}\) then \(\sabs{f(b)-f(a)} \leq M \sabs{b-a}\text{.}\)

11.1.9.

Prove that there is no simple mean value theorem for integrals for complex-valued functions: Find a continuous function \(f \colon [0,1] \to \C\) such that \(\int_0^1 f = 0\) but \(f(t) \not= 0\) for all \(t \in [0,1]\text{.}\)

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