Tagged Questions

10
votes
2answers
197 views

Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?

Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable? Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
4
votes
4answers
101 views

what is the relation of smooth compact supported funtions and real analytic function?

What is the major difference between real analytic and test function (smooth compact supported functions). Can we find a real analytic function $f$ on $R^n$ which is also smooth compact supported? If ...
-3
votes
0answers
38 views

Give an example of a complete analytic function.

Give an example of a complete analytic function for which over a point a = 0 is the branching order of 2 and 4, as well as a significant feature.Exactly I dont know how to do it.We know just we have: ...
3
votes
0answers
60 views

How do zeros on the complex plane affect the real number line?

Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
1
vote
2answers
51 views

Weierstrass factorization of sine, and related questions

So the idea is that you can represent a function as a product of its zeroes, and there are some fundamental factors that often crop up. I am interested in, give this is the WF of sine : Is it ...
1
vote
0answers
31 views

Relations between complex functions satisfying a specific condition

What is the relation between the following two complex functions: $$g(\theta)=\sum_n x[n]\ y[n]\ e^{in\theta}$$ and $$f(\theta)=\sum_n \left(x[n]\pm i\sqrt{1-x[n]^2}\right)\ y[n]\ e^{in\theta}$$ ...
0
votes
3answers
89 views

How to show that $\frac{z-a}{a-z}$ has not inverse?

How to show that $\frac{z-a}{a-z}$ has not inverse? I know that $$\frac{z-a}{a-z}=\frac{-1(a-z)}{a-z}=-1 .$$ But state if I'm wrong, that following is true: $$f(z)=\frac{z-a}{a-z} \Leftrightarrow ...
0
votes
1answer
166 views

Finding the Range of a Complex Function

I am taking Complex Analysis right now and having difficulty understanding how to find the range of complex functions, it seems that there is no standard way to do it and that each problem is ...
1
vote
1answer
276 views

problem on complex analysis

Let $U$ be an open subset of $C$ containing $D=\{ z\in \Bbb C :|z|\leqslant 1\}$ and let $f\colon U\rightarrow C$ defined by $f(z)=e^{i\theta}\frac {z-a}{1-\overline az}$ for $a\in D$ and ...
2
votes
1answer
74 views

Complex logarithm and injectivity

Please forgive the trivial nature of this question: let U be a connected domain inside the punctured unit disk so that every curve inside it has winding number zero around the origin. Is the complex ...
2
votes
1answer
201 views

Is there a geometric projection for every complex function

I was wondering about the best way to visualize complex functions. As they're $$ R^2 \rightarrow R^2$$ i think best way are complex plane image/grid transforms like they used in the Dimensions movie ...
1
vote
2answers
189 views

Complex analytic functions and their zeros.

I am having trouble with the following statement found in a textbook: "Let $U$ be a connected open set. Let $f$ be a complex analytic function on $U$ and not constant. Either $f$ is locally ...
0
votes
1answer
119 views

Finding the Residue of a Holomorphic Function

I'm trying to find the residue at $0$ of the function $$f(z)=\frac{1+iz-e^{iz}}{z^3}$$ on $\mathbb{C} - \{0\}$. I think it's a double pole at the origin, but I'm not entirely sure. I'm wondering if ...
1
vote
0answers
207 views

Using Rouche's theorem

Let $p>1$. Consider $\phi(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^pdt$. Function $\phi(p)$ is analytic on its domain. It's derivative, $\phi'(p)=\int_0^{\infty}\left|\frac{\sin ...
2
votes
3answers
516 views

Why is there no continuous square root function on $\mathbb{C}$?

I know that what taking square roots for reals, we can choose the standard square root in such a way that the square root function is continuous, with respect to the metric. Why is that not the case ...
1
vote
0answers
189 views

How can I find the Laurent series of $f(z) = \ln(1+\exp(z))$ about its singularities?

I think my problem below can be solved by finding the Laurent series of $f(z) = \ln(1+\exp(z))$ about its points of singularities. (Any better suggestion is more than welcome!) How might I find such a ...
7
votes
3answers
327 views

Why is there no continuous log function on $\mathbb{C}\setminus\{0\}$?

Over the years, I've often heard that there is no logarithm function which is continuous on $\mathbb{C}\setminus\{0\}$. The usual explanation is usually some handwavey argument about following such a ...
2
votes
1answer
196 views

Image of a map in the complex plane

Is there an elegant way (either intuitive/ by a series of diagrams or by manipulating numbers/algebra) to find out what the image of $\sin(w)$ where $w\in \mathbb C$ from a domain say $\{w\in \mathbb ...
0
votes
1answer
150 views

Harmonic conjugate

I have been asked the following question and would appreciate an explanation. Suppose we have to find an analytic function $F(z)$ where $z=x+iy\in \mathbb C$ and its real part is $g(x,y)$. Question: ...
4
votes
0answers
514 views

Finding general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$.

I'm trying to find the most general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$. I write this polynomial as $u(x,y)$. I calculate $$ \frac{\partial^2 u}{\partial x^2}=6ax+2by,\qquad ...
0
votes
1answer
332 views

If $\operatorname{Re}^{2}(x)=-1$, what is $x$?

$i=\sqrt{-1}$ $\operatorname{Re}(z)+i\cdot\operatorname{Im}(z)=z$ If $\operatorname{Re}^{2}(x)=-1$, what is $x$? $x$ cannot be defined in complex number as $(a+ib)$. { $a$ and $b$ are real numbers ...
1
vote
1answer
57 views

Complex Analysis: Correspodence between $H(\Omega_1)$ and $H(\Omega_2)$.

If we let $$\Omega_1=\left\{z \in \mathbb{C} : 0<\operatorname{Im}(z)< \pi\}, \quad \Omega_2=\{z \in \mathbb{C}: 0<\operatorname{Im}(z) \right\},$$ can we establish a one-to-one ...
1
vote
2answers
532 views

Holomorphic and Harmonic functions

I'm studying holomorphic functions in my Complex Analysis class and have encountered the following problem: Let $U,V$ be open subsets of $\mathbb{C}$ and $f$ be a holomorphic function on $U$ with ...
0
votes
1answer
160 views

Analytic function Cauchy-Riemann with initial condition

I have some trouble with the initial condition of this equation: $f(z) = u(x+iy)+iv(x+iy)$ where $u(x+iy)=x^2-y^2+x$ and initial condition $f(i)=-1+i$. Can someone please help me how to come to ...
0
votes
0answers
55 views

graph of the size of a complex function

Here there are two graphs for two functions from $R^2\mapsto R$. Is there similar graph for the absolute value of an analytic complex variable function $f:C\mapsto C$ that has the same point (like ...
1
vote
1answer
238 views

Complex Analysis: Cauchy-Riemann Problem

I'm working through a problem set in Complex Analysis and have encountered the following question: Problem Write down the Cauchy-Riemann equations and explain their connection with ...
6
votes
1answer
166 views

Complex Analysis: Continuity of Function

Problem Define g to be the function $g(z)=re^{\frac{i\theta}{2}}$ if $z=re^{i\theta}$ with $r>0$ and $-\pi<\theta\le\pi$, and $g(z)=0$ when $z=0$. Is $g$ continuous from ...
1
vote
0answers
37 views

Lower bounds for holomorphic functions on annuli with explicit bounds on their power series

Let $f$ be a holomorphic function on $\mathbf{C}$ and consider its restriction to the annulus $X=B(0,1) - B(0,3/4)$ in the complex plane. (Here $B(0,r)$ is the open disc with radius $r$ around $0$.) ...
1
vote
1answer
288 views

Extending the cube root function to $\mathbb{C}$

On $\mathbb{R}$, the cube-root function (call it $f(x)$) is a well-defined single-valued function which is $C^\infty$ except at the origin. ($f(27)=3$, $f(-27)=-3$, etc.) The complex cube root (call ...
4
votes
0answers
236 views

Organizing types of functions by their calculus-related properties, in diagram form?

Does anyone know of a diagram that displays and organizes categories of functions according to their calculus-related properties (e.g. continuous, $C^\infty$, degrees of differentiability and ...
-3
votes
3answers
388 views

How do I find out the symmetry of a function?

For example, how do I know that with: $$f(x_1,x_2,x_3,x_4)=\frac{x_1 x_2+x_3 x_4-x_2 x_3-x_1 x_4}{x_1 x_2+x_3 x_4-x_1 x_3-x_2 x_4}$$ $f$ has the property: ...
1
vote
1answer
218 views

transformation matrices and complex functions as projections

This question is about the connection between linear algebra and complex analysis. Coming from a two real dimensional domain a transformation matrix geometrically transforms a set of points (e.g. a ...
1
vote
2answers
248 views

A stereographic projection related question

This might be an easy question, but I haven't been able to up come up with a solution. The image of the map $$f : \mathbb{R} \to \mathbb{R}^2, a \mapsto (\frac{2a}{a^2+1}, \frac{a^2-1}{a^2+1})$$ is ...