Recent questions in Analyzing functions

Analyzing functions
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roncadort8d
2022-12-17

I have two questions:

Question 1

For Functions with a holomorphic numerator and polynomial denominator, can I simply find the Taylor Series of the numerator in the neighbourhood of the first Singularity, and determine the type in the neighbourhood? And do so for each singularity?

Example

Given $f(z)=\frac{{e}^{z}}{{z}^{2}+1}$ with Singularities at $z=\pm i$. Let us check the type of z=i first. Since ${e}^{z}\in H(\mathbb{C})$, we can use the taylor expansion in the neighbourhood of i, ie. ${B}_{\u03f5}(i)$ where $0<\u03f5<2$ so that $-i\notin {B}_{\u03f5}(i)$. Then

$f(z)=\frac{1}{(z-i)(z+i)}\sum _{k=0}^{\mathrm{\infty}}\frac{{e}^{i}}{k!}(z-i{)}^{k}$

$=\frac{1}{z+i}\sum _{k=0}^{\mathrm{\infty}}\frac{{e}^{i}}{k!}(z-i{)}^{k-1}$

$=\frac{1}{z+i}(\frac{{e}^{i}}{z-i}+\sum _{k=1}^{\mathrm{\infty}}\frac{{e}^{i}}{k!}(z-i{)}^{k-1})$

So i has a Pole of first order. Similarily we can show that −i also is a Pole of first order.

Is that right?

Question 2 How do you go about functions like $f(z)=\frac{1-{e}^{z}}{1+{e}^{z}},|z|<4$?

I've found the Singularities, namely $z=\pm i\pi $, but have no idea how to proceed, because I can't find a way to use a similar approach to the above example...!

Analyzing functions
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compagnia04
2021-12-25

minimum - ?

maximum - ?

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William Burnett
2021-12-14

find the following.

(a) Use the limit definition of the derivative to find

f(x)=?

(b) Find the equation of the tangent line at

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impresijuzj
2021-11-30

Find the ares of the rezion completely enclosed by the graphs of the given functions fand g.

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Serotoninl7
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siroticuvj
2021-11-29

following four functions. Only one of the functions has

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Analyzing functions
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rescuedbyhimw0
2021-11-28

Analyzing functions
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Sheelmgal1p
2021-11-28

Analyzing functions and their corresponding graphs can be a daunting task, however with the right knowledge it can be quite straightforward. When analyzing a function, you can determine its domain and range, the x- and y-intercepts, the maximum and minimum points, the intervals of increase and decrease, the intervals of concavity, the end behavior, and the asymptotes. Additionally, when analyzing rational functions, you can find the vertical and horizontal asymptotes, the x- and y-intercepts, and the holes in the graph. Understanding and analyzing functions and their graphs is an important part of any math course, but don't worry if you're having trouble understanding. We have questions that might to help you with questions and equations, and to provide you with answers.