Question

Residue of \(\tan z\) at \(z=\pi / 2\) is \(\ldots \ldots\)

Short answer

The residue of \( \tan z \) at \( z = \pi / 2 \) is \( -1 \).

Step-by-step solution

Identify the Function and its Behavior

The function under consideration is \( \tan z = \frac{\sin z}{\cos z} \). \( \tan z \) has simple poles where \( \cos z = 0 \). One such pole is at \( z = \frac{\pi}{2} \). Our task is to find the residue of \( \tan z \) at this point.

Recognition of Pole Order

Since \( \cos z \) has a simple zero at \( z = \frac{\pi}{2} \) (i.e., zero of order 1), it follows that \( \tan z \) has a simple pole at this point. Therefore, we can use techniques for residues at simple poles.

Compute the Residue at a Simple Pole

For a function \( f(z) = \frac{g(z)}{h(z)} \) with a simple pole at \( z = z_0 \) due to \( h(z_0) = 0 \), the residue is given by \( \text{Res}(f, z_0) = \frac{g(z_0)}{h'(z_0)} \). Here, \( g(z) = \sin z \) and \( h(z) = \cos z \).

Find the Derivative of the Denominator

Compute \( h'(z) = \frac{d}{dz} \cos z = -\sin z \). Evaluate at the pole, \( h'\left(\frac{\pi}{2}\right) = -\sin\left(\frac{\pi}{2}\right) = -1 \).

Evaluate the Residue

The residue of \( \tan z = \frac{\sin z}{\cos z} \) at \( z = \frac{\pi}{2} \) is \( \frac{\sin\left(\frac{\pi}{2}\right)}{-1} = \frac{1}{-1} = -1 \).

Conclusion

Thus, the residue of \( \tan z \) at \( z = \frac{\pi}{2} \) is \( -1 \).

Key concepts

Residue Calculation

Residue calculation is a fundamental concept in complex analysis. It involves finding the residue of a complex function at a particular point, often a pole, within a given domain.
A residue is essentially the coefficient of the \( \frac{1}{z-z_0} \) term in the Laurent series expansion of the function about a point \( z_0 \).

• Residues are crucial in evaluating complex integrals. By using the Residue Theorem, many integral computations are simplified into finding these residues.
• In our example, finding the residue of \( \tan z \) at \( z = \frac{\pi}{2} \) helps determine the integral of this function around that point.

Understanding residue calculations can immensely ease solving complex analysis problems.

Simple Poles

A simple pole is a type of singularity in a complex function. It occurs where the function behaves like \( \frac{1}{z-z_0} \) near a point \( z_0 \). At a simple pole, a function \( f(z) \) can be expressed as \[ f(z) = \frac{g(z)}{h(z)} \] where \( h(z_0) = 0 \) and \( h'(z_0) eq 0 \).

• Unlike poles of higher order, at a simple pole only a single term of degree -1 exists in the function's Laurent series.
• For \( \tan z = \frac{\sin z}{\cos z} \), \( z = \frac{\pi}{2} \) is a simple pole because \( \cos z = 0 \) and \( \sin z \) does not zero out at that point.

Determining simple poles simplifies calculating residues, using \[ \text{Res}(f, z_0) = \frac{g(z_0)}{h'(z_0)} \].

Analytic Functions

Analytic functions, or holomorphic functions, are functions that are both complex differentiable and have derivatives that are continuous in a given domain. In the complex plane, these functions show a surprising degree of smoothness.
This makes them crucial in complex analysis since their behavior can be controlled perfectly if boundaries are set around their domains of analyticity.

• Key features of analytic functions include having power series expansions at points within their domain.
• If a function is analytic except for isolated singularities, tools like the Residue Theorem can be applied.

In the example of \( \tan z \), its behavior is analytic everywhere except where it has poles, such as \( z = \frac{\pi}{2} \). More importantly, residues are only needed at these points.

Trigonometric Functions in Complex Plane

Trigonometric functions can be extended into the complex plane, such as \( \sin z \) and \( \cos z \). These functions behave differently in the complex plane compared to the real. They come with periodicity, and identities known from real numbers also hold when variables are complex.

• In complex analysis, each trigonometric function has a periodic set of singularities or zeros.
• For \( \tan z = \frac{\sin z}{\cos z} \), zeros of cosine correspond to simple poles of tangent, like at \( z = \frac{\pi}{2} \).
• Through Euler's formula, trigonometric functions can also be expressed using exponential functions, which is useful for calculations in complex domains.

The trigonometric functions' properties govern their poles and zeros and are vital for computations like residue finding.