Recent questions in Analyzing functions

Differential CalculusAnswered question

roncadort8d 2022-12-17

Types of Singularities for singularities of the complex function $\frac{1-{e}^{z}}{1+{e}^{z}}$

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...!

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...!

Differential CalculusAnswered question

compagnia04 2021-12-25

Find the absolute extrema of the function on the closed interval.

$y=3{x}^{\frac{2}{3}}-2x,\text{}[-1,\text{}1]$

minimum - ?

maximum - ?

minimum - ?

maximum - ?

Differential CalculusAnswered question

Oberlaudacu 2021-12-23

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

$y=\frac{{e}^{x}}{x};\text{}(1,{e}^{x})$

Differential CalculusAnswered question

Danelle Albright 2021-12-21

Find the length of the curve.

$\overrightarrow{r}\left(t\right)=<8t,{t}^{2},\frac{1}{12}{t}^{3}>,0\le t\le 1$

Differential CalculusAnswered question

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 $f{}^{\u2033}\left(2\right)$

Differential CalculusAnswered question

Joan Thompson 2021-12-20

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

Differential CalculusAnswered question

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$x=5$ .

find the following.

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

f(x)=?

(b) Find the equation of the tangent line at

Differential CalculusAnswered question

hvacwk 2021-12-11

Find the area enclosed by the curve $x={t}^{2}-2t,\text{}y=\sqrt{t}$ and the $y-a\xi s$

Differential CalculusAnswered question

Donald Johnson 2021-12-09

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

$f(x,y)=xy;\text{}c\left(t\right)=(\mathrm{cos}\left(t\right),\text{}\mathrm{sin}\left(t\right));\text{}0\le t\le 2\pi$

Differential CalculusAnswered question

Monincbh 2021-12-03

Differential CalculusAnswered question

Susan Munoz 2021-12-02

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

$f(x,y)={x}^{2}{y}^{2};\text{}c\left(t\right)=(\mathrm{cos}\left(t\right),\text{}3\mathrm{sin}\left(t\right));\text{}0\le t\le 2\pi$

Differential CalculusAnswered question

impresijuzj 2021-11-30

Sketch the graphs of the functions.

$f\left(x\right)={x}^{3}+{x}^{2}-2x\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}g\left(x\right)=0$

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

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

Differential CalculusAnswered question

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.

$h\left(x\right)=3-\frac{1}{2}x$

Differential CalculusAnswered question

Serotoninl7 2021-11-30

Determine which functions are polynomial functions. For those that are, identify the degree.

$f\left(x\right)=\frac{{x}^{2}+7}{3}$

Differential CalculusAnswered question

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.

$f\left(x\right)={e}^{x}$

following four functions. Only one of the functions has

critical numbers. Which is it? Explain.

Differential CalculusAnswered question

rescuedbyhimw0 2021-11-28

Derivatives involving other trigonometric functions Find the derivative of the following functions.

$y=\frac{\mathrm{tan}w}{1+\mathrm{tan}w}$

Differential CalculusAnswered question

Sheelmgal1p 2021-11-28

Can the functions be added, subtracted, multiplied, and divided except where the denominator is zero to
produce new functions?

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.