Tagged Questions
10
votes
2answers
190 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 $f^3$ ...
0
votes
1answer
43 views
Prove that the taylor series of cos(z) and sin(z) are holomorphic
I have a question on an old exam paper given as follows:
If we define
$$\cos(z) = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - ... \frac{z^{2n}}{(2n)!} + ... = \sum_{n=0}^{\infty} ...
1
vote
2answers
58 views
“Differentiable in the sense of real analysis”
What does it mean for a complex function to be 'differentiable in the sense of real analysis'.
I understand that a function if complex differentiable, if and only if it is 'differentiable in the ...
1
vote
4answers
73 views
Calculating a complex derivative of a polynomial
What are the rules for derivatives with respect to $z$ and $\bar{z}$ in polynomials?
For instance, is it justified to calculate the partial derivatives of ...
0
votes
0answers
40 views
Analyticity and differentiability
May this is be an easy question. I have some problem with difference between consepts Analyticity and differentiability a complex function.
0
votes
1answer
49 views
Is there a function which has an analytic region but the derivatives on the region is not continuous?
Cauchy's theorem state if $f$ is analytic in a region $R$ and on its boundary $C$ then, $f(a)=1/2πi∮f(z)/(z-a)dz$. It was first proved by Green's theorem with added restriction that $f$ has a ...
2
votes
2answers
59 views
estimating the derivatives of $\sin z$ at $z=0$
(a) Use Cauchy inequality to obtain estimate for the derivatives of $\sin z$ at $z=0$ and
(b) determine how good these estimates are
No examples are given except the proof on Cauchy Inequality. ...
1
vote
1answer
42 views
To show that a partial dertivative (of a piecewise function) is continuous at $0$
$$f(z)=\cases{\frac{x^4-6x^2y^2+y^4}{x^2+y^2}
+i\frac{4xy(x^2-y^2)}{x^2+y^2},& $z\ne0$\cr 0, &$z=0$}$$
Let $u=\Re(f)$.
I have shown from first principles that $\frac{\partial ...
0
votes
1answer
141 views
Show that this piecewise function is differentiable at $0$
I have shown (from first principles) that the Cauchy-Riemann equations for the following function are satisfied at $z=0$. But to properly prove differentiability at $z=0$, what should I do next? Do I ...
2
votes
2answers
37 views
Holomorphic function $f$ such that $f'(z_0) \neq 0$
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an holomorphic function such that $f'(z_0) \neq 0$ for some $z_0 \in \mathbb{C}$.Prove that there is $r>0$ such that, if $|z-z_0|<r$ and $z \neq z_0 ...
4
votes
1answer
99 views
If $f$ is analytic where $f$ is represented as $f=g.h$ where $g$ is analytic . From here can we conclude that $h$ is analytic?
If $f$ is analytic, where $f$ is represented as $f=g \cdot h,$ where $g$ is analytic. From here can we conclude that $h$ is analytic?
3
votes
1answer
106 views
If$f(z)$ is analytic , then what about $f'(z)?? $
If$f(z)$ is analytic , then what about $f'(z)$?
can we conclude that $f^{(k)}(z)$ is analytic for any k$\in $$ \mathbb{N} $
0
votes
1answer
47 views
A change of variables in the euler equation
If someone could help me with the proposed change of variables, it would be greatly appreciated. Consider Euler's equation:
$$z^2w'' + \alpha zw' + \beta w = 0$$
where $w$ is a function of $z$ and ...
1
vote
3answers
52 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
0answers
55 views
Uniform convergence of complex exponent derivative
I'm trying to prove the following:
Let $\Re z > 0$. Then $$\lim_{\varepsilon \to 0} \frac{t^{z + \varepsilon} - t^z}{\varepsilon} = t^z \log t$$ uniformly in $t \in [0,1]$.
I've tried to ...
6
votes
3answers
206 views
Is the complex derivative “speed”?
The first thing I was told about the real derivative is that it's "how fast the function is growing" at a given point. This interpretation wasn't addressed in my complex analysis classes. Can the ...
7
votes
4answers
194 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:
$$f'(z)=\lim_{\Delta z\to ...
2
votes
1answer
63 views
Derivative of the Selberg $\zeta$-function
I want to compute the derivative of the Selberg $\zeta$-function:
$$ \mathcal{Z}(s)=\prod_{\gamma \; \text{primitive}} \prod_{n=0}^\infty (1-e^{-l(\gamma)(n+s)}); \qquad \Re(s)>1.$$
Where ...
1
vote
0answers
43 views
$ g\left( z \right) = \int\limits_\gamma {\frac{{\varphi \left( u \right)}} {{u - z}}du} $ computing the derivate
Let $\gamma\colon[a,b]\to \mathbb{C}$ denote a piecewise differentiable path , and let $\varphi:$ Image $\gamma\colon \to \mathbb{C}$ be a continuous function.
Define $g: D = ...
3
votes
1answer
78 views
Derivative of order N of a product of complex functions
Given the function:
$$F=\frac{\exp(-iTz)}{z}\prod_{k=1}^N(1-\frac{z}{z_k})^{-1}$$
with $z\in C$
is it possible to give a closed expression of the $n^{th}$ derivative of $F$ with respect to $z$?:
...
0
votes
1answer
62 views
a question about the derivate with respect to z , of a composition
This is a problem from a book that I'm using to study complex analysis. I'm a little insecure
with what I have to prove here, because, I don't know what it means $g_w$ for example . I'm a little ...
0
votes
2answers
84 views
Derivative of a particular function
Under what conditions is the derivative of
$$f(z) = (z-(x_1+iy_1))(z-(x_2+iy_2))(z-(x_3+iy_3))$$ equal to $3(z^2-13)$ where $i$ is the imaginary number? When I put the equation in Wolfram it's a huge ...
1
vote
0answers
207 views
Using Rouche's theorem
Let $p>1$.
Consider $\phi(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^pdt$.
Function $\phi(p)$ is analytic on its domain.
It's derivative, $\phi'(p)=\int_0^{\infty}\left|\frac{\sin ...
2
votes
1answer
364 views
Differentiating a complex function
How would u differentiate this function w.r.t. z -
$$\frac{1}{z-2+3i}$$
U would need to split it and get partial derivatives right? Although im not sure how you'd split it into real and imaginary ...
1
vote
1answer
124 views
Branch points of rational functions
Let $f$ be a rational function on a compact connected Riemann surface $X$. The rational function $f$ induces a holomorphic map $\overline{f}:X\to \mathbf{P}^1(\mathbf{C})$.
Let $x$ be a point on the ...
0
votes
1answer
351 views
derivative of absolute value of a complex function
If $f:U\subset\mathbb{C}\mapsto\mathbb{C}$, where $f(x+iy)=u(x,y)+iv(x,y)$ is a meromorphic function and if $f$, $f'$, and $f''$ are not zero in the strip $a<x<b$, can we get ...
