The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...
1
vote
0answers
33 views
How to compute $\int_0^\infty \frac{\sin t}{t^{s+1}} dt $?
How to compute $\displaystyle\int_0^\infty \frac{\sin t}{t^{s+1}}\;\text dt$ ?
Here, the real part of the complex number $s$ is negative and greater than $-1$.
3
votes
0answers
47 views
zeros of exponential polynomials
Let $\exp[n;z]$ denote the nth Taylor polynomial for
the exponential function.
In the 1920's Szego initiated the study of the asymptotic
properties of the zeros (rescaled by dividing by $n$) of
this ...
-1
votes
2answers
37 views
complex analysis -examples of complex functions that are bounded and the limits
What are some examples of complex functions that are bounded and the limits does not exists as $z\to 0$?
0
votes
1answer
26 views
complex analysis-roots of constant polynomial function
True or False: There exists a value for $w\in \mathbb{C}$ such that the equation $z^{100}+z +1 = w$ has no solution with $z\in\mathbb{C}$.
My answer
$True$-By fundamental theorem of algebra,Every ...
2
votes
2answers
54 views
Contour Integrals and Residues
I'm trying to figure out what it is all about, but my mind is blowing up. First of all, I have turned back and looked at the general definitions of integrals. Then I have looked to line integrals. ...
2
votes
1answer
58 views
Why is Cauchy's integral formula always written with the function as the subject?
Why is Cauchy's integral formula always written as $$f(w)=\frac1{2\pi i}\int_L\frac{f(z)}{z-w}dz$$ instead of as $$\int_L\frac{f(z)}{z-w}dz=2\pi i f(w)$$? Isn't the latter form how it's typically ...
0
votes
0answers
40 views
Radius of convergence and complex power series
I recall from calculus that the radius of convergence of a power series is the same for the derivative and the integral of that power series.
For complex, however, is this the same for derivatives ...
0
votes
0answers
31 views
On using fourier transforms to solve the root of a convolution
In continuation of Lower bounds of laplace transform of characteristic functions.
My question is:
Can anyone point out where i'm going wrong in the derivation below.
It's been a while ...
-2
votes
2answers
76 views
Complex Analysis boundedness and limits
True or false. If $f: \mathbb{C}\to\mathbb{C}$ is bounded, then $\lim_{z\to 0} f(z)$ exists.
R If f is bounded means that there is some M∈R such that ∀z ∈C holds that |f(z)|≤M.
My Answer
Would ...
0
votes
0answers
26 views
Is the intersection of compact Stein sets, a compact Stein set?
Brian Conrad, in an article of his, defined a compact Stein set to be a compact set (subset of a complex manifold) K admitting Hausdorff neighborhood such that H^i(K,G)=0 for all i>0, and for all G ...
1
vote
1answer
40 views
$f_n(z)={z^n\over n}$, $z\in D$ open unit disk then
$f_n(z)={z^n\over n}$, $z\in D$ open unit disk then
1.$\sum f_n$ converges uniformly on $D$?
2.$f_n$ and $f'_n$ converges uniformly on $D$?
3.$\sum f'_n$ converges on $D$ pointwise?
4.$f_n''(z)$ ...
-1
votes
0answers
38 views
1 complex addition = 2 real additions, 1 complex multiplication = 4 multiplications + 2 additions.
Could you show that, when talking about Fourier transforms, that one complex addition requires 2 real additions, and one complex multiplication requires 4 multiplications and 2 additions.
0
votes
0answers
9 views
harmonic, m-harmonic, hyperbolic-harmonic
Anyone has any idea about harmonic, m-harmonic, and hyperbolic-harmonic functions?
Harmonic functions are characterized by mean value property, m-harmonic functions are characterized by volume mean ...
0
votes
0answers
23 views
uniformization theorem - squares and circles
I am trying to understand the uniformization theorem and get some intuition about it.
The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of ...
1
vote
1answer
51 views
Integral sign with circle (AND arrow on the circle) through it
I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path.
So when I encountered in complex analysis the above integral sign but with ...
1
vote
1answer
41 views
Is there a text book containing a self-contained and complete proof of the Jordan Curve theorem?
I seem to remember (in my undergraduate years) encountering a book on complex analysis which contained a proof of the Jordan Curve Theorem, building up from first principles - so self-contained and ...
2
votes
1answer
73 views
Simple conceptual Fundamental Thm of Calculus question
When applying the Fundamental Thm of Calculus in complex analysis, what does it mean for an open connected set to contain a loop? For example, does my red-color open annulus contain the black colour ...
0
votes
0answers
36 views
Quaternion exponential map, rotations and interpolation
A code snippet I need to optimize is performing something peculiar. It seems that it's somehow related to transforming from a frame of reference to another. This is what it does, in mathematical ...
0
votes
3answers
34 views
Continuity of the real and imaginary parts of a continuus complex-valued function
If a complex-valued function is continuous, are the component real and imaginary parts $u(x,y)$ and $u(x,y)$ necessarily continuous? If so, why?
2
votes
3answers
44 views
Cauchy's Theorem- Trigonometric application
any help would be very much appreciated. The question asks to evaluate the given integral using Cauchy's formula. I plugged in the formulas for $\sin$ and $\cos$ ($\sin= \frac{1}{2i}(z-1/z)$ and ...
0
votes
1answer
42 views
Is a series (summation) of continuous functions automatically continuous?
I'm being asked to show that a given series (of rational functions) converges uniformly on a given disc, and then and asked to use this fact to show that integrating its limit function (i.e. a ...
2
votes
2answers
60 views
Rouché Theorem to calculate the number of zeros
How can I calculate the number of zeros of $\cos z+3z^3$ using the Rouché Theorem?
0
votes
0answers
35 views
Spectrum of a unitary
I have a unitary element $v$ in $C(S^{1}, \mathbb{C})$ with full spectrum (the whole circle). Is it possible to construct another unitary $u$ in $C(S^{1}, \mathbb{C})$ out of $v$ such that the ...
4
votes
2answers
48 views
Does the Weierstrass M-test show analyticity?
I'm trying to show (textbook exercise) that the riemann-zeta function is analytic. The solution is here:
Why does the proof say that the zeta series converges to an analytic function? Doesn't the ...
1
vote
2answers
36 views
Is it possible for a function to be analytic anywhere outside the circle of convergence of its power series expansion?
Is it possible for a function to be analytic anywhere outside the circle of convergence of its power series expansion? I'm referring to analytic fuctions of course (i.e. those with power series ...
1
vote
1answer
44 views
Is $e^z\sum_{k=0}^\infty\frac{k^3}{3^k}z^k$ analytic inside $|z|=3$?
Am I correct that the following function is analytic at least inside $|z|=3$? (I used the ratio test.) The solutions manual says that the function is analytic on and inside |z|=1, so I wonder if I'm ...
-2
votes
1answer
30 views
Two problems on analytic function and Mapping of elementary functions
Let $G$ be a region and let $f$ and $g$ be analytic functions on $G$ such that $f(z)g(z)=0$ for all $z \in G$. Show that either $f$ or $g$ is identically zero on $G$.
Here is how I do it: Assume $f$ ...
4
votes
1answer
50 views
number of zeros of function $\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)-1$
$$f(z)=\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)-1$$
How many zeros does the above function have in $\Bbb{C}$?
6
votes
2answers
67 views
Is a curve homologous to zero according to Ahlfors actually homologous to zero?
The presentation of the homology version of Cauchy's theorem in Ahlfors is slick, but sweeps a lot of the topology under the rug using clever arguments. This question is an attempt to reconcile ...
2
votes
2answers
119 views
Diagonalizability in $\mathbb{R}$ and $\mathbb{C}$
Give an example of a matrix $A\in M_{n\times n}(\mathbb{R})$ that is not diagonalizable, but A is diagonalizable viewed as a matrix over the field of complex numbers $\mathbb{C}.$
2
votes
1answer
63 views
Why do injective holomorphic functions have nonzero derivative
For some open sets $U$, $V$ in the complex plane, let $f:U\rightarrow V$ be an injective holomorphic function. Then $f'(z) \ne 0$ for $z \in U$.
Now I don't understand the proof, but here it is ...
1
vote
1answer
22 views
Mapping of a Lens-shaped region by a Möbius Transformation
Consider the 'lens' described by $\{z:|z-i|<\sqrt{2}\ \text{and}\ |z+i|<\sqrt{2} \}$ . We want to map this to the upper right quadrant using a Möbius transformation.
The two circles meet at ...
0
votes
0answers
54 views
Vanishing of Dirichlet Series
Suppose the function
$\sum_{n=1}^{\infty}{a_{n}n^{-s}}$ is $0$ on some open set $U\subset\mathbb{C}$. (Can assume the sum converges absolutely on $U$.)
Is it true that $a_{n}=0$ for all $n$?
(This ...
2
votes
2answers
53 views
Constructing a conformal map from $\mathbb{D}$ to a cut plane
Source: Oxford Exam $A2 \ 1999$
We want to construct a conformal map $F$ from the unit disc $\mathbb{D}=\{z:|z|<1\}$ to $\mathbb{C} \setminus S$ where $S$ is the half-line $\{x+i:x \in (-\infty,0] ...
0
votes
0answers
27 views
Harmonic Function Transformation Help
Consider the harmonic function $$u(x,y)=1-y+\frac{x}{x^2+y^2}$$ on the upper half plane $y>0$.
What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
0
votes
2answers
50 views
Complex number question
For any complex numbers $z_1, z_2$, is the quantity $S$: $$
S = 4 \left(| z_1|^6 + |z_2 |^6\right ) + 4 |z_1|^3 |z_2 |^3 + \left(2 |z_1|^2\times \overline{z_1}^2\times z_2^2\right) + \left(2 ...
3
votes
2answers
83 views
Plotting in the Complex Plane
I just wonder how do you plot a function on the complex plane? For example,$$f(z)=\left|\dfrac{1}{z}\right|$$
What is the difference plotting this function in the complex plane or real plane?
Thank ...
1
vote
2answers
79 views
Must a complex power series *fail* to be convergent somewhere on its circle of convergence?
My textbook asserts so, but I can't find its proof of the claim. On the other hand, a lecture slide I'm cross-referencing claims that a power series is allowed to be convergent at ALL points of its ...
2
votes
0answers
32 views
show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle [duplicate]
I need to show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle except the point $z=1$, well at $z=1$ we get our known divergent harmonic series, but I am not able to show easily ...
1
vote
2answers
34 views
Understanding the (Partial) Converse to Cauchy-Riemann
We have that for a function $f$ defined on some open subset $U \subset \mathbb{C}$ then the following if true:
Suppose $u=\mathrm{Re}(f), v=\mathrm{Im}(f)$ and that all partial derivatives ...
3
votes
1answer
35 views
Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$
I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i ...
10
votes
4answers
341 views
Why is $2\pi i \neq 0?$ [duplicate]
We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$
Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's ...
10
votes
1answer
115 views
How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?
I need to solve the to following integral:
$$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx.$$
I tried this integral in Mathematica, but it was not able to solve it. ...
1
vote
3answers
40 views
Laurent expansion problem
Expand the function $$f(z)=\frac{z^2 -2z +5}{(z-2)(z^2+1)} $$ on the ring $$ 1 < |z| < 2 $$
I used partial fractions to get the following $$f(z)=\frac{1}{(z-2)} +\frac{-2}{(z^2+1)} $$
then
...
3
votes
1answer
35 views
Application of the Identity Theorem to $|x|^3$ for $-1<x<1$
Oxford Exam $2602$ $1997$ $Q3$
We want to show that there is no function $f$ which is holomorphic in $D(0;1)$ and such that $f(x)=|x|^3$ for $-1<x<1$.
Here are my thoughts thus far:
Suppose ...
1
vote
1answer
21 views
Results following from Analyticity on a domain
This is part of an old Oxford exam paper (1997 2602 Q2) I'm working on for revision.
Suppose we have a function $f$ which is holomorphic on the disc radius $R$ about $0$. We want to show that there ...
2
votes
2answers
47 views
Integrate: $\int_0^{\pi} \log ( 1 - 2 r \cos \theta + r^2)d\theta$
If $r \in \Bbb R$ how to integrate $\displaystyle \int_0^{\pi} \log ( 1 - 2 r \cos \theta + r^2)d\theta$?
I need some hints. Special case, if $r = 1$ then I know the above integral is zero.
Here ...
1
vote
3answers
34 views
Series of $\int_0^z \zeta^{-1} \sin \zeta d \zeta$
This is a homeworkquestion so I would appreciate some good hints. I have $f(z) = \int_0^z \zeta^{-1} \sin \zeta d \zeta$. Can this be written as a power-series in $\mathbb C$ around $z = 0$?
1
vote
2answers
49 views
having trouble intuiting analyticity
My textbook seems to suggest that the analytic functions are precisely the functions that can be written in terms of $z$ alone (no $x$ or $y$ or conjugate-$z$).
Am I inferring correctly?
Does this ...
3
votes
1answer
72 views
Integrate: $\int_0^1 \frac{1}{\sqrt[3]{x^2 - x^3}}dx$
How to integrate using Residue theorem.
$$\int_0^1 \frac{1}{\sqrt[3]{x^2 - x^3}}dx$$
How do I choose my branch-cut particularly? I was reading this article on wikiepdia and I think it is related. ...
