Question

Let \(h(z)\) be analytic and have a simple zero at \(z=z_{0}\), and let \(g(z)\) be analytic there. Let \(f(z)=g(z) / h(z)\), and show that $$ \operatorname{Res}\left[f\left(z_{0}\right)\right]=\frac{g\left(z_{0}\right)}{h^{\prime}\left(z_{0}\right)} $$.

Short answer

The residue of the function \(f(z) = g(z) / h(z)\) at the simple zero \(z_{0}\) of \(h(z)\) can be shown to be \(\frac{g(z_{0})}{h'(z_{0})}\), using the definition of residue and applying Cauchy's integral formula.

Step-by-step solution

Recall Definition of Residue

The residue of a function \(f(z)\) at a point \(c\) is given by \(\operatorname{Res}[f(z);c] = \frac{1}{2\pi i} \oint f(z) dz\), with the integral taken over a small anti-clockwise contour around the point \(c\). In this case, we need to compute this for our function \(f(z) = g(z) / h(z)\) at the point \(z_{0}\) where \(h(z_{0}) = 0\), which is a simple zero.

Express Integral in terms of \(g(z)\) and \(h(z)\)

The residue is given by the integral of the function around \(z_{0}\). We can express this as \(\operatorname{Res}[f(z_{0})] = \frac{1}{2\pi i} \oint \frac{g(z)}{h(z)} dz\). Since \(z_{0}\) is a simple zero, we can write \(h(z)\) as \((z - z_{0})h_1(z)\), where \(h_1(z_{0}) \neq 0\). Thus the integral becomes \(\frac{1}{2\pi i} \oint \frac{g(z)}{(z - z_{0})h_1(z)} dz\).

Apply Cauchy's Integral Formula

Now, applying Cauchy's integral formula, which states that for \(f(z)\) analytical around \(z_{0}\), the integral over a loop containing \(z_{0}\) is \(2 \pi i f(z_{0})\), we can see that the integral above will give \(\frac{g(z_{0})}{h_1(z_{0})}\).

Replace \(h_1(z)\) with its definition

Recalling that \(h(z)\) = \((z - z_{0})h_1(z)\), we differentiate this to get \(h'(z)\) = \(h_1(z) + (z - z_{0})h_1'(z)\). However, evaluating at \(z_{0}\) we get \(h'(z_{0}) = h_1(z_{0})\), since \((z_{0} - z_{0}) = 0\). Thus \(\frac{g(z_{0})}{h_1(z_{0})}\) can be written as \(\frac{g(z_{0})}{h'(z_{0})}\).

Key concepts

Analytic Functions

Analytic functions, also known as holomorphic functions, play a central role in complex analysis. An analytic function is a complex function that is locally given by a convergent power series. This means that around any point within its domain, an analytic function can be expressed as a power series.

An essential property of analytic functions is that they are infinitely differentiable, and their derivatives are also analytic. This differentiability is not just a local property; it extends throughout the domain of the function.

Examples

Common examples of analytic functions include basic trigonometric functions, exponential functions, and polynomials when considered in the complex plane.

Importance in Residue Calculus

In the context of residue calculus, if a function is analytic at a point and in its vicinity except for isolated singularities, it is possible to apply various powerful results, like Cauchy's integral formula, to evaluate integrals around those singularities.

Cauchy's Integral Formula

Cauchy's integral formula is a fundamental theorem in complex analysis which provides a very direct way of computing integrals of analytic functions over closed curves. This formula states that if you have an analytic function defined on and inside a simple closed contour, then the value of the integral of that function over the curve is directly related to the function's value at any point inside the curve.

Formula Statement

More precisely, if \(f(z)\) is analytic inside and on a simple closed contour \(C\) and \(a\) is any point inside \(C\), then: \[ \oint_C \frac{f(z)}{z - a} dz = 2\pi i f(a). \] This result is exceptionally useful because it allows the evaluation of integrals in terms of the values that the function takes on, rather than having to perform complex integration tasks.

Connection with Residues

Cauchy's integral formula is closely related to the residue theorem, which leverages residues at singular points to compute contour integrals. It demonstrates the powerful impact singularities have on the value of a contour integral.

Simple Zero

In the realm of complex analysis, a zero is a point where a given complex function equals zero. A simple zero is a type of zero where the function touches the x-axis only once before changing direction. Mathematically, \(f(z_0) = 0\) defines a zero at \(z_0\), and it is considered 'simple' if the first derivative at that point \(f'(z_0)\) is non-zero.

Properties of Simple Zeros

One significant aspect of simple zeros is that they do not repeat — the function does not stay at zero for an interval but rather crosses the x-axis. That means the function changes from positive to negative or vice versa, indicating a sign change at the zero.

Role in Residue Calculation

In residue calculus, the residue of a function at a simple zero is of great interest because it simplifies the calculation of the integral around a pole. As illustrated in the proposed exercise, understanding the behavior of the function near its zeros allows for the computation of residues and, by extension, the evaluation of complex integrals.