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Chapter 8: Problem 2

For each of the following, find the principal part at the pole given: a) \(\frac{z^{2}+3 z+1}{z^{4}}(z=0)\) b) \(\frac{z^{2}-2}{z(z+1)}(z=0)\) c) \(\frac{e^{z} \sin z}{(z-1)^{2}}(z=1)\) d) \(\frac{1}{z^{2}\left(z^{3}+z+1\right)}(z=0)\)

Short Answer

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Question: Determine the principal part of each function at the given pole. a) \(\frac{z^{2}+3 z+1}{z^{4}}\) at z=0 b) \(\frac{z^{2}-2}{z(z+1)}\) at z=0 c) \(\frac{e^{z}\sin z}{(z-1)^{2}}\) at z=1 d) \(\frac{1}{z^{2}\left(z^{3}+z+1\right)}\) at z=0 Answer: a) \(\frac{3}{z^3}\) b) \(\frac{-2}{z}\) c) 0 d) \(\frac{1}{z}\)

Step by step solution

01

Determine the residue at the given pole for each function

For simple poles, we can find the residue of the function by: Residue = limit as z approaches to pole of (z-a) times the function. For other poles, we need to expand the function using the Laurent series and identify the coefficients of the negative powers of (z-a). a) \(\frac{z^{2}+3 z+1}{z^{4}}(z=0)\) This function has a pole of order 4 at z=0. The residue at the pole z=0 is the coefficient of the \(\frac{1}{z^3}\) term in the Laurent series expansion. Since the Laurent series expansion of this function is: $$\frac{z^2+3z+1}{z^4} = z^{-4} + 3z^{-3} + z^{-2}$$ The residue is 3, so the principal part is \(\frac{3}{z^3}.\) b) \(\frac{z^{2}-2}{z(z+1)}(z=0)\) This function has a simple pole at z=0. Let us find the residue using the following formula: Residue = \(\lim_{z\to 0}(z\times \frac{z^2-2}{z(z+1)})\) Residue = \(\lim_{z\to 0} \frac{z^2-2}{z+1}\) = -2 The principal part is \(\frac{-2}{z}\) c) \(\frac{e^{z}\sin z}{(z-1)^{2}}(z=1)\) This function has a pole of order 2 at z=1. Let us first expand the function using Laurent series around z=1: $$\frac{e^z\sin z}{(z-1)^2}=\frac{e^{1+z-1}\sin (1+(z-1))}{(z-1)^2}=\frac{e\sin (1+\frac{z-1}{1!}+\frac{-(z-1)^2}{2!}+\cdots)}{(z-1)^2}$$ Now, expand the numerator using Taylor series at z=1: $$e\sin(1) + e\cos(1)(z-1) + \frac{e\sin(1)-(z-1)e\cos(1)}{2}(z-1)^2 + \cdots$$ Divide this by \((z-1)^2\) to get the Laurent series around z=1: $$\frac{e\sin(1)-(z-1)e\cos(1)}{2} + e\cos(1)+O\left((z-1)^{-1} \right)$$ The principal part is the terms containing negative powers of \((z-1)\). In this case, there is no such term, thus the principal part is 0. d) \(\frac{1}{z^{2}\left(z^{3}+z+1\right)}(z=0)\) This function has a pole of order 2 at z=0. Let us find the residue using the following formula: Residue = limit as z approaches to pole of z times the function. Residue = \(\lim_{z\to 0}z\frac{1}{z^2(z^3+z+1)}\) To find this limit, we'll expand the function using the Taylor series at z=0: $$\frac{1}{z^{2}\left(z^{3}+z+1\right)}=\frac{1}{z^2}\frac{1}{z^3 + z +1}=\frac{1}{z^2}\frac{1}{1 + O(z)}$$ Now, divide this by z to get the limit: $$\lim_{z\to 0} z\frac{1}{z^2(1+O(z))}=\frac{1}{z\left(1+O(z)\right)}$$ The principal part is \(\frac{1}{z}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Residue Calculus
Residue calculus is a powerful tool in complex analysis used to evaluate integrals of complex functions around a given pole. Essentially, the residue of a function at a point provides information about the behavior of that function near a pole. A pole exists at point \(z = a\) if the function goes to infinity as \(z\) approaches \(a\). Residue calculus often involves calculating residues at these poles to simplify computations of complex integrals.
  • The residue is the coefficient of \(\frac{1}{z-a}\) in the Laurent series expansion of the function around the pole \(z = a\).
  • For simple poles, residues can often be found by multiplying the function by \(z-a\) and taking the limit as \(z\) approaches \(a\).
  • For poles of higher order, the function is typically expanded using Laurent series and the residue is identified as the term that contains \(\frac{1}{z-a}\).
Residue calculus is particularly useful in evaluating integrals over closed contours in the complex plane using the Residue Theorem. This theorem relates the contour integral of a function to the sum of residues of the function's poles inside the contour.
Laurent Series
The Laurent series is a representation of a complex function as an infinite series. Unlike the Taylor series, which includes only non-negative powers of \((z-a)\), the Laurent series can include both positive and negative powers. This quality allows it to represent functions near singularities.
  • The function \(f(z)\) can be expressed as:\[f(z) = \sum_{n=-\infty}^{\infty} a_n (z-a)^n\]where \(a_n\) are the coefficients, and \(z = a\) is known as the singularity or point of expansion.
  • The principal part of the Laurent series is the sum of the terms with negative powers:
    • \[\text{Principal Part} = \sum_{n=-\infty}^{-1} a_n (z-a)^n\]
  • The coefficients of these terms provide important information about the poles and residues.
Laurent series are particularly used when the function has a pole, making them crucial in the study of complex functions around singular points.
Complex Analysis
Complex analysis is a branch of mathematics that studies functions of complex numbers. It provides powerful techniques and tools for solving a variety of problems, particularly those involving integration and differentiation of functions that are not easily solved by real analysis alone.
  • Complex functions are often expressed in terms of their behavior in the complex plane, encompassing both magnitude and direction.
  • Key concepts in complex analysis include analytic functions, which are differentiable everywhere in their domain, and meromorphic functions, which are analytic except at their poles.
  • Residue calculus, Laurent series expansions, and the study of poles and their proper manipulation are fundamental topics within complex analysis.
Complex analysis not only assists in theoretical mathematics but also finds applications in physics, engineering, and applied mathematics, where complex potentials and fields are often analyzed.
Poles of Functions
Poles are a type of singularity in functions where they exhibit an undefined behavior, typically going to infinity as the variable approaches a certain point. Understanding poles is essential for performing accurate calculations in complex analysis.
  • A function \(f(z)\) is said to have a pole of order \(n\) at \(z=a\) if \((z-a)^n f(z)\) is analytic and non-zero at \(z=a\).
  • Pole order signifies how far into the principal part the power series expansion of the function will extend.
  • The residue at the pole is derived from the coefficient of \(\frac{1}{z-a}\) in the Laurent series, emphasizing why identifying poles contributes directly to finding residues.
  • Simple poles, of order 1, are the most straightforward to work with, while higher-order poles require more calculation, often through series expansion.
By dissecting the nature of poles and leveraging tools like Laurent series, complex analysis provides deeper insights into the behavior of functions near singular points.

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Most popular questions from this chapter

Evaluate the following integrals: a) \(\int_{0}^{1+i}\left(x^{2}-i y^{2}\right) d z\) on the straight line from 0 to \(1+i\). b) \(\int_{0}^{\pi} z d z\) on the curve \(y=\sin x\). c) \(\int_{1}^{1+i} \frac{d z}{2}\) on the line \(x=1\).

Show that each branch of \(\log z\) is analytic in each domain in which \(\theta\) varies continuously and that $$ (d / d z) \log z=1 / z . $$ [Hint: Show from the equations \(x=r \cos \theta, y=r \sin \theta\) that \(\partial \theta / \partial x=-y / r^{2}, \partial \theta / \partial y=\) \(x / r^{2}\). Show that the Cauchy-Riemann equations hold for \(u=\log r, v=\theta\).]

Prove the formula (8.43). [Hint: If \(w=\sin ^{-1} z\), then \(2 i z=e^{i w}-e^{-i w}\); multiply by \(e^{i w}\) and solve the resulting equation as a quadratic for \(e^{i w}\).]

a) Let \(w=f(z)\) be a linear fractional transformation mapping \(|z| \leq 1\) on \(|w| \leq 1\). Prove that \(f(z)\) has the form (8.98). [Hint: Let \(z_{0}\) map on \(w=0\). By Problem \(\left.10 \mathrm{~b}\right) 1 / \bar{z}_{0}\) must map on \(w=\infty\). The point \(z=1\) must map on a point \(w=e^{i \beta}\). Take \(z_{1}=1, z_{2}=z_{0}\). \(z_{3}=1 / z_{0}\) in \(\left.9 \mathrm{c}\right)\).] b) Prove that every transformation (8.98) maps \(|z| \leq 1\) on \(|w| \leq 1\).

Determine the radius of convergence of each of the following serics: a) \(\sum_{n=0}^{\infty} \frac{z^{n}}{n^{2}+1}\) b) \(\sum_{n=0}^{\infty} \frac{(z-1)^{n}}{3^{n}}\) c) \(\sum_{n=0}^{\infty} n ! z^{n}\) d) \(\sum_{n=0}^{\infty} \frac{z^{\prime \prime}}{n !}\)
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