Tagged Questions
3
votes
3answers
103 views
Limit on a spiral
I was thinking about limits of functions along various spirals and this one stumped me a bit. The limit that needs to be found is ultimately:
$$\lim_{\varphi\to\infty} \coth\varphi\csc\varphi$$
...
1
vote
1answer
20 views
Existence of limit of a function of a complex variable.
Say that I have a function $f(z)=f(x+iy)=f(x,y)$ and I want to investigate whether the limit at a particular point, say $z=0$ exists.
I recall that within the domain of real numbers, I checked ...
1
vote
1answer
63 views
Check my work: $\lim a_n = 0 \Rightarrow \lim \sqrt{a_n} = 0 $? (for $a_n$ positive)
I'm trying to prove, as "properly" as possible the following:$$\left[ \lim z_n = z \right] \iff \left[ \lim x_n = x \quad \wedge \quad \lim y_n = y \right]$$
where $z_n = x_n + i y_n$ and $z=x+iy$.
...
0
votes
1answer
32 views
how to show uniform convergence for sequence $u_n(z) = n z e^{-nz^2}$ such that $\Re[z^2] > 0$
How to show uniform convergence for $u_n(z) = n z e^{-nz^2}$ such that $\Re[z^2] > 0$
Here is my attempt letting $z = x + iy$:
$$
\begin{align*}
|u_n - 0| &= |nz e^{-nz^2}| \\
&\le n ...
3
votes
1answer
109 views
Can't prove this limit of complex numbers from a paper
Okay so I found in a paper, marked as "simple exercise", the following thing: for $z,b \in \mathbb{C}$,
$$\lim_{b\to0} \frac{1}{z-b} + \frac{1}{2b} - ...
0
votes
0answers
58 views
Is it possible to find the length of a limit cycle in advance?
Let $z$ be a complex number. Let $n$ be a positive integer. Let $f(z)$ be a given entire transcendental function.
Let $I$ be a given complex number. By definition $F_0=I,F_1=f(I),F_n=f(F_{n-1})$.
A ...
1
vote
1answer
60 views
Plessner's theorem and radial limit of derivative of univalent function
I'm reading an article by Pommerenke (Conformal mapping and linear measure) and I have one question about it. First the assumptions:
Let $f$ be analytic and univalent in the unit disk $D$ and let ...
1
vote
1answer
57 views
Behavior at infinity.
Classify the behavior at $\infty$ for $$f(z)=\frac{\sin z}{z^2},\,g(z)=\frac{1}{\sin z},\,h(z)=\exp\left(\tan\frac{1}{z}\right).$$
So I just considered $f(1/z),g(1/z),h(1/z)$ at $z=0$. For $f$ I ...
1
vote
2answers
56 views
Find limit as $y\to \infty$
Find the limit of $$\frac{-(iy - 1/2) -|iy-1/2|}{(iy - 1/2) -|iy-1/2|}$$ as $y\to \infty$.
The answer is $i$ but I dont know how to show step by step.
if i divide by y on top and bottom, I get
...
0
votes
1answer
43 views
$\displaystyle\lim_{z\to3i}\frac{-3z}{z^2-9-6iz}$
Is the following limit $\infty$ or does it not exist?
I'm asking this question to hopefully dispel a confusion I'm having (Limit of $f(z)=\frac 1z$ as $z$ approaches $0$?).
...
3
votes
3answers
94 views
Limit of $f(z)=\frac 1z$ as $z$ approaches $0$?
I know that the answer is infinity, but doesn't this contradict the theorem stating that $\displaystyle\lim_{z\to0} \Im(f(z)) = \Im(\displaystyle\lim_{z\to0} f(z))$? I mean, the LHS doesn't exist ...
0
votes
2answers
99 views
Constructing Taylor series
I am having trouble constructing the answer to this problem, which is also linked here: Convergence Properties of the Taylor Series for $\frac{1+z}{1-z}$.
Find and state the convergence properties ...
1
vote
2answers
73 views
Elementary proof of convergence
Prove that if the sequence $\{z_n\}_{n=1}^{\infty}$ converges, then
$(z_n-z_{n-1})\to 0$ as $n\to\infty$.
My attempt:
Suppose $\text{lim}_{n\to\infty}z_n= L$, then $\exists$ $\epsilon >0$ ...
1
vote
1answer
73 views
Radius of convergence power series
Find the radius of convergence of each of the following power series:
a.) $\sum_{j=1}^{\infty}\frac{2^j}{3^j+4^j}z^j$
b.) $\sum_{j=0}^{\infty}2^jz^{j^2}$
c.) ...
1
vote
1answer
44 views
Simple question on limit of the modulus
I know that for real functions, if the limit at infinity of $f$ is a number $l$, then the limit of $|f|$ is $|l|$. Is this true for complex functions too? Thanks.
I'm thinking that if a function ...
0
votes
1answer
41 views
Maclaurin expansions
Let $f(z)=\sum_{j=0}^{\infty}a_jz^{j}$ be the Maclaurin expansion of
a function $f(z)$ analytic at the origin. Prove each of the following:
a.) $\sum^{\infty}_{j=0}a_jz^{2j}$ is the ...
1
vote
1answer
52 views
Elementary converge and diverge
Why does the geometric series $\sum^{\infty}_{j=0}c^j$ converge when $|c|<1$, but diverge when $|c|\ge 1$? Since the geometric series is $= \frac{1}{1-c}$, which means it is undefined at $c=1$, but ...
1
vote
3answers
36 views
How can I find $\lim_{|z|\to 0}|\frac{1-e^{2iz}}{z}|$?
In calculation of a contour integral, I need to find
$$\lim_{|z|\to 0}\bigg|\frac{1-e^{2iz}}{z}\bigg|.$$
Let $z=re^{i\theta}$. Then
$$
\lim_{|z|\to ...
1
vote
2answers
56 views
Explain why the residue is equal to the limit?
I'm studying for a midterm and my teacher warned this would be a good question to understand for the test. The problem is, I do not know how to go about explaining it.
Suppose g(x) has a pole of ...
2
votes
1answer
100 views
Integral with Undefined Endpoint (Complex Variables)
The problem is find: $\int\limits_0^1 \lim\limits_{n\rightarrow\infty}(nz^{n-1})dz$
I started by finding $\lim\limits_{n\rightarrow\infty}(nz^{n-1})$. Naturally it converges to zero on [0,1). However ...
0
votes
3answers
73 views
Proof of $\displaystyle \lim_{z\to 1-i}[x+i(2x+y)]=1+i$
I am having some difficulty with the epsilon-delta proof of the limit above.
I know that $|x+i(2x+y)-(1+i)|<\epsilon$ when $|x+iy-(1-i)|<\delta$.
I tried splitting up the expressions above in ...
1
vote
1answer
49 views
Limit of $u(x,y),v(x,y)$ at $0$ where $f(z)=u+iv$
I have an exercise with the following function
$$f(z)=\begin{cases}
\frac{z^{5}}{|z|^{4}} & z\neq0\\
0 & z=0
\end{cases}$$
I have prove that Cauchy-Riemann equations are satisfied at ...
0
votes
1answer
50 views
Extended functions continuous at $z=(0,0)$
There are 4 functions:
$$\frac{Re(z)}{|z|},\frac{z}{|z|},\frac{Re(z^2)}{|z|^2},\frac{zRe(z)}{|z|}$$
I need to determine which of these functions can be defined at $z=0$ in such a way that the ...
1
vote
3answers
53 views
Why is it clear from this formulation that f is continuous wherever it is holomorphic?
Hi I am new on here so not sure if this is right place to post but quick and presumably easy question:
So holomorphic at a point $z_0 \in \Omega$ is defined as the limit as $h\rightarrow 0$ of
...
1
vote
1answer
47 views
Calculating a limit with constraints
Given the function $f(x)$,
$$ f(x,y,z,w) = \frac{x+iy}{\sqrt{|w+z|}} \text{.} $$
How do I calculate the limit
$$ \lim\limits_{w\rightarrow -z} f $$
under the constraint that the points $(x,y,z) ...
5
votes
2answers
146 views
how to show that $\lim_{z \to 0}z^z$ does not exist?
What makes $0^0$ indeterminate. Here is a video by numberphile that claims that $z^z$ does not exist as $z \to 0$ where $z \in \mathbb C $. I tried tried $\lim_{x \to 0}(x+ix)^{(x+ix)}$ and replaced ...
0
votes
1answer
106 views
Limits of complex numbers
"We say $z_n \rightarrow \infty$ if, for each positive number $M$ (no
matter how large), there is an integer $N$ such that $|z_n |>M$
whenever $n > N$; similary $lim_{z\rightarrow ...
2
votes
2answers
103 views
Why is this function entire? $f(z) = z^{-1} \sin{z} \exp(i tz)$
In problem 10.44 of Real & Complex Analysis, the author says $f(z) = z^{-1} \sin{z} \exp(i tz)$ is entire without explaining why. My guess is that $z = 0$ is a removable singularity, $f(z) = 1$ ...
6
votes
1answer
79 views
What is $\lim\limits_{z \to \pi/2} \tan^2(z) $ for $z \in \mathbb C$?
I am trying to evaluate the following limit ($z \in \mathbb C$):
$$\lim\limits_{z \to \pi/2} \tan^2(z) $$
I get the following solution:
$$\lim\limits_{x \to \pi/2} \tan^2(x) = \lim\limits_{x \to ...
2
votes
2answers
134 views
What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C$?
What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C^*$? I need it to determine the type of the singularity at $z = 0$.
5
votes
2answers
88 views
Computing $\lim_{s \to 1} \Gamma \left(\frac{1-s}{2}\right) (s-1)$
I want to evaluate the following limit: $$\lim_{s \to 1}\; \Gamma \left( \frac{1-s}{2} \right) (s-1).$$
I know that the gamma function has simple poles at $-n$ for $n \in \mathbb{N}_0$ with residue ...
2
votes
1answer
65 views
Limit $\lim_{n\to\infty}\frac{\exp(ia_1)+\exp(ia_2)+…+\exp(ia_n)}{n}=\alpha$
Show that for any sequence $a_1,a_2,...$ of real numbers, the two conditions
$\lim_{n\to\infty}\frac{\exp(ia_1)+\exp(ia_2)+...+\exp(ia_n)}{n}=\alpha$
and
...
0
votes
0answers
75 views
Residue at an integration border in case of a limit?
I am dealing with an integral in a limit of the following shape:
$$\lim_{\epsilon \to 0} \int_0^{\frac{\pi}{2}} dx \frac{2 \epsilon}{1-(1-\epsilon^2)\sin^2(x)}$$
Formally, assuming that ...
4
votes
2answers
526 views
Help with an irregular integral
I am looking for help with doing the following integral :
$$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in ...
3
votes
1answer
76 views
Calculating the Value of a complex limit
I am given some limits that exist, I'm supposed to find their values. Seems really simple however I am struggling.
Find the value of $\displaystyle \lim_{z\to\\i}\frac{z^4 - 1}{z-i} $.
My ...
2
votes
0answers
36 views
If $y(z) = C y_0(z) \int_w^z \frac{d\zeta}{y_0(\zeta)^2}$, what limit can we take in $C$ and $w$ to obtain $y(z) \to y_0(z)$?
This is Exercise 6.5 from Miller's Applied Asymptotic Analysis.
The book shows that, given any solution $y_0(z)$ to the equation
$$
y''(z)+f(z)y(z)=0,
\tag{1}
$$
a general solution is given by
$$
...
1
vote
2answers
88 views
Find the limit of $\frac{\bar{z}}{z}$ as $z$ goes to $0$.
I put it in exponential form to get $\dfrac{re^{-i \theta}}{re^{i \theta}}$ but I think I'll get $\frac{0}{0}$ which isn't defined and isn't a good enough proof to say it doesn't have a limit.
0
votes
1answer
84 views
Complex differentiability of $C^1$ function of one complex variable
Assuming that $f$ has $C^1$ real and imaginary parts, prove that:
$$\displaystyle \lim_{r \to 0} \frac{1}{r^2} \int_{C_r}f(z)dz = 2\pi i\frac{\partial{f}}{\partial{\bar{z}}}(z_0).$$
Additionally, ...
7
votes
4answers
199 views
How do I find $\frac{\text{d}}{\text{d}z}\left(z\bar{z}\right)$?
I am seeking $\frac{\text{d}}{\text{d}z}\left(z\bar{z}\right)$ where $f(z)=z\bar{z}.$
And I know that I need to use the following definition of the derivative:
$$f'(z)=\lim_{\Delta z\to ...
2
votes
1answer
125 views
Does $\lim_{z\rightarrow0}\frac{\bar{z}^2}{z^2}$ exits?
For$$\lim_{z\rightarrow0}\frac{\bar{z}^2}{z^2}$$
Does $\lim_{z\rightarrow0}\frac{\bar{z}^2}{z^2}$ exits? If the answer is no, why?
Does $\bar{z}$ represents $a-bi$?
1
vote
2answers
357 views
Complex function limit
I asked time ago about the limit of a complex number $z$ over its conjugate, as $z$ goes to infinity.
Now I have a strategy, it is to convert $z$ to trigonometric form, and the limit depends uniquely ...
3
votes
4answers
461 views
Limit of a complex number over its conjugate, as z approach the infinity.
this question must be pretty easy but i´m just taking my firs course in complex variables, i need to find the limit of Z over it conjugate as z reach the infinity. I need to know how to do it in a ...
3
votes
2answers
179 views
Multiple-choice question regarding $\lim\limits_{n \to \infty} \sum\limits_{k = 1}^n \left| e^{2\pi ik/n} − e^{2\pi i(k−1)/n} \right|$
The limit
$$\lim_{n \to \infty} \sum_{k = 1}^n \left| e^{2\pi ik/n} − e^{2\pi i(k−1)/n} \right|$$ is
(A) $2$
(B) $2e$
(C) $2\pi$
(D) $2i$.
I can't solve ...
4
votes
1answer
103 views
A simple question : Is $g(z + \Delta z).g(z) = g(z)^2$?
Is $g(z + \Delta z).g(z) = g(z)^2$ ?
The full expression is a $\lim_{\Delta z \to 0} \frac {A}{G} $, where $G$ is the left-hand side of the above expression.
It's a question about a solution to a ...
3
votes
0answers
160 views
What is $i^{i^{\,{}^{.\,^{.\,^{.\,^i}}}}}$ equal to? [duplicate]
Possible Duplicate:
How to compute the infinite tower of the complex number $i$, that is$ ^{\infty}i$
Define $a_0 = i$, and $a_n = i^{a_{n-1}}$.
What is
$$\lim_{n \rightarrow ...
0
votes
2answers
172 views
limit of $conj(z)/z$, $z\rightarrow 0$
I just watched a video lecture where it's proved that $\lim_{z\rightarrow 0}{\frac{\bar{z}}{z}}$ does not exist: by putting the condition that $z=x+iy$ and $y=\lambda x$, it comes out that ...
0
votes
2answers
102 views
Absolute convergence for all values except the values $z=\left( 1+\frac {a} {m}\right) e^{\frac {2k\pi i} {m}}$
I am trying to show that the series $$\sum _{n=1}^{\infty }\dfrac {nz^{n-1}\left( \left( 1+\dfrac {1} {n}\right) ^{n}-1\right) } {\left( z^{n}-1\right) \left( z^{n}-\left( 1+\dfrac {1} {n}\right) ...
1
vote
0answers
145 views
Limit of exp(z) along a half-line
I'm still new to complex functions and not very confident yet, so I wonder if you all can check if I've understood this problem correctly:
Suppose we take the limit of of $e^z$ as $|z| \to \infty$ ...
2
votes
2answers
328 views
Limits at infinity
$f(z), g(z)$ are two entire functions, both have no zeros in the closed upper half plane. What does it mean/imply that $$\bigg| \lim_{y\rightarrow \infty}\frac{f(z)}{g(z)}\bigg|=c$$
($z=x+iy$)
i.e. ...
3
votes
1answer
133 views
Question concerning proof of the Cauchy Integral Formula on the Wolfram website
I was reading through the proof of the Cauchy Integral Formula here. I do not understand
how the transition is made from equation (8) to equation (9). While taking the limit as $r\to 0$, doesn't the ...
