# Learn Analyzing Functions with These Examples

Recent questions in Analyzing functions

## Types of Singularities for singularities of the complex function $\frac{1-{e}^{z}}{1+{e}^{z}}$I have two questions:Question 1For 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?ExampleGiven $f\left(z\right)=\frac{{e}^{z}}{{z}^{2}+1}$ with Singularities at $z=±i$. Let us check the type of z=i first. Since ${e}^{z}\in H\left(\mathbb{C}\right)$, we can use the taylor expansion in the neighbourhood of i, ie. ${B}_{ϵ}\left(i\right)$ where $0<ϵ<2$ so that $-i\notin {B}_{ϵ}\left(i\right)$. Then$f\left(z\right)=\frac{1}{\left(z-i\right)\left(z+i\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{e}^{i}}{k!}\left(z-i{\right)}^{k}$$=\frac{1}{z+i}\sum _{k=0}^{\mathrm{\infty }}\frac{{e}^{i}}{k!}\left(z-i{\right)}^{k-1}$$=\frac{1}{z+i}\left(\frac{{e}^{i}}{z-i}+\sum _{k=1}^{\mathrm{\infty }}\frac{{e}^{i}}{k!}\left(z-i{\right)}^{k-1}\right)$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 $f\left(z\right)=\frac{1-{e}^{z}}{1+{e}^{z}},|z|<4$?I've found the Singularities, namely $z=±i\pi$, but have no idea how to proceed, because I can't find a way to use a similar approach to the above example...!

zabuheljz 2022-08-18

## Find $\frac{dy}{dx}$ if $3{y}^{4}+{x}^{7}=5x$

compagnia04 2021-12-25

## Find the absolute extrema of the function on the closed interval. minimum - ? maximum - ?

Oberlaudacu 2021-12-23

## Find an equation of the tangent line to the given curve at thespecified point

Danelle Albright 2021-12-21

## Find the length of the curve. $\stackrel{\to }{r}\left(t\right)=<8t,{t}^{2},\frac{1}{12}{t}^{3}>,0\le t\le 1$

Carla Murphy 2021-12-21

## If $f\left(2\right)=10$ and ${f}^{\prime }\left(x\right)={x}^{2}f\left(x\right)$ for all x, find $f{}^{″}\left(2\right)$

Joan Thompson 2021-12-20

## Find c such that fave $=f\left(c\right),f\left(x\right)={\left(x-3\right)}^{2},\left[2,5\right]$

William Burnett 2021-12-14

## Use the function $f\left(x\right)=\frac{9}{x-4}$ find the following. (a) Use the limit definition of the derivative to find $f\left(x\right)$. f(x)=? (b) Find the equation of the tangent line at $x=5$.

hvacwk 2021-12-11

## Find the area enclosed by the curve and the $y-a\xi s$

Donald Johnson 2021-12-09

## Find the maximum and minimum values attained by the function falong the path $c\left(t\right)$.

Monincbh 2021-12-03

## $1.{\int }_{\left\{}^{\left\{}\mathrm{sin}\frac{2}{3}xdx$ $2.{\int }_{\left\{}^{\left\{}x\mathrm{cos}6{x}^{2}dx$ $3.{\int }_{\left\{}^{\left\{}\mathrm{tan}9xdx$ $4,{\int }_{\left\{}^{\left\{}{x}^{2}{\mathrm{cot}x}^{3}dx$

Susan Munoz 2021-12-02

## Find the maximum and minimum values attained by the function falong the path $c\left(t\right)$.

impresijuzj 2021-11-30

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

zachutnat4o 2021-11-30

## Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. $h\left(x\right)=3-\frac{1}{2}x$

Serotoninl7 2021-11-30

## Determine which functions are polynomial functions. For those that are, identify the degree. $f\left(x\right)=\frac{{x}^{2}+7}{3}$

siroticuvj 2021-11-29

## Critical Numbers Use a graphing utility to graph the following four functions. Only one of the functions has critical numbers. Which is it? Explain. $f\left(x\right)={e}^{x}$

rescuedbyhimw0 2021-11-28

## Derivatives involving other trigonometric functions Find the derivative of the following functions. $y=\frac{\mathrm{tan}w}{1+\mathrm{tan}w}$

Sheelmgal1p 2021-11-28