Chapter 23, Series Representations of Functions Video Solutions, Advanced Engineering Mathematics, International Student Edition

Problem 1

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$\cos (2 z) ; z=0$

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Problem 2

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$e^{-z} ; z=-3 i$

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Problem 3

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$\frac{1}{1-z} ; 4 i$

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In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$\sin \left(z^2\right) ; 0$

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Problem 5

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$\frac{1}{(1-z)^2} ; 0$

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Problem 6

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$\frac{1}{2+z} ; 1-8 i$

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Problem 7

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$z^2-3 z+i ; 2-i$

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Problem 8

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$1+\frac{1}{2+z^2} ; i$

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Problem 9

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$(z-9)^2 ; 1+i$

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Problem 10

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$e^z-i \sin (z) ; 0$

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Problem 11

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$\sin (z+i) ;-i$

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Problem 12

In each of Problems 1 through 12, find the Taylor series of the function about the point. Also determine the radius of convergence and open disk of convergence of the series.
$\frac{3}{z-4 i} ;-5$

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01:12

Problem 13

Suppose $f$ is differentiable in an open disk about zero, and satisfies $f^{\prime \prime}(z)=2 f(z)+1$. Suppose $f(0)=1$ and $f^{\prime}(0)=i$. Find the Maclaurin expansion of $f(z)$.

Ryan Williams

Numerade Educator

01:33

Problem 14

Find the first three terms of the Maclaurin expansion of $\sin ^2(z)$ in three ways as follows:
(a) First, compute the Taylor coefficients at 0 .
(b) Find the first three terms of the product of the Maclaurin series for $\sin (z)$ with itself.
(c) Write $\sin ^2(z)$ in terms of the exponential function and use the Maclaurin expansion of this function.