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Chapter 10: Problem 37

Expand \(\sinh z\) in a Taylor series about the point \(z=i \pi\).

Short Answer

Expert verified
The Taylor series expansion of \(\sinh(z)\) about the point \(z=i \pi\) is \(-1*(z-i \pi) + 1*(z-i \pi)^3 - 1*(z-i \pi)^5 + 1*(z-i \pi)^7 - ...\)

Step by step solution

01

Evaluate function at the point of interest

Determine the value of \(\sinh(z)\) at \(z=i \pi\). The hyperbolic sine function is given by \(\sinh(z) = \frac{e^z - e^{-z}}{2}\). Hence, \(\sinh(i \pi) = \frac{e^{i \pi} - e^{-i \pi}}{2} = 0\).
02

Evaluate first derivative at the point of interest

Determine the first derivative of \(\sinh(z)\) at \(z=i \pi\). The derivative of \(\sinh(z)\) is \(\cosh(z)\), which is \(\frac{e^z + e^{-z}}{2}\). Hence, \(\cosh(i \pi) = \frac{e^{i \pi} + e^{-i \pi}}{2} = -1\).
03

Evaluate second derivative at the point of interest

Determine the second derivative of \(\sinh(z)\) at \(z=i \pi\). The derivative of \(\cosh(z)\) is again \(\sinh(z)\). Hence, \(\sinh''(i \pi) = \sinh(i \pi) = 0\).
04

Compute and summarize the Taylor series

Given the results from Steps 1 to 3, and knowing that the pattern of derivatives of \(\sinh(z)\) at \(z=i \pi\) will continue in the series, the Taylor series expansion of \(\sinh(z)\) around \(z=i \pi\) is: \(0 + -1*(z-i \pi) + 0*(z-i \pi)^2 + 1*(z-i \pi)^3 - ...\).

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Key Concepts

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

Hyperbolic Functions
Hyperbolic functions, similar to trigonometric functions, are a set of functions that have important applications in various branches of mathematics, including algebra, geometry, calculus, and particularly in solving problems involving hyperbolic geometry and complex analysis. Two fundamental hyperbolic functions are the hyperbolic sine ( ( ( ( ( ( ( 鈥� 鈥� 鈥� 鈥� 鈥� 聽聽聽聽聽聽聽聽聽聽聽聽聽鈥� 聽 聽 聽 聽 聽 聽 聽 聽 聽 聽 聽 聽 聽 聽 聽 )

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Most popular questions from this chapter

Let \(f\) be analytic within and on the circle \(\gamma_{0}\) given by \(\left|z-z_{0}\right|=r_{0}\) and integrated in the positive sense. Show that Cauchy's inequality holds: $$\left|f^{(n)}\left(z_{0}\right)\right| \leq \frac{n ! M}{r_{0}^{n}}$$ where \(M\) is the maximum value of \(|f(z)|\) on \(\gamma_{0}\).

Show that \(\left|e^{-z}\right|<1\) if and only if \(\operatorname{Re}(z)>0\).

Let \(f(t)=u(t)+i v(t)\) be a (piecewise) continuous complex-valued function of a real variable \(t\) defined in the interval \(a \leq t \leq b\). Show that if \(F(t)=U(t)+i V(t)\) is a function such that \(d F / d t=f(t)\), then $$\int_{a}^{b} f(t) d t=F(b)-F(a) .$$ This is the fundamental theorem of calculus for complex variables.

Obtain the first few nonzero terms of the Laurent series expansion of each of the following functions about the origin. Also find the integral of the function along a small simple closed contour encircling the origin. (a) \(\frac{1}{\sin z}\); (b) \(\frac{1}{1-\cos z}\); (c) \(\frac{z}{1-\cosh z}\); (d) \(\frac{z^{2}}{z-\sin z}\); (e) \(\frac{z^{4}}{6 z+z^{3}-6 \sinh z}\); (f) \(\frac{1}{z^{2} \sin z}\); (g) \(\frac{1}{e^{z}-1}\).

Show that (a) \(\tanh \left(\frac{z}{2}\right)=\frac{\sinh x+i \sin y}{\cosh x+\cos y}\), (b) \(\operatorname{coth}\left(\frac{z}{2}\right)=\frac{\sinh x-i \sin y}{\cosh x-\cos y}\).
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