0
votes
4answers
63 views

Integral of a function $f:\mathbb{R}\rightarrow \mathbb{C}$

My real analysis book defines derivatives and integrals only for a function $f:A\rightarrow \mathbb{R}$, where $A\subset \mathbb{R}$. But, when talking about Fourier series, it comes out an integral ...
0
votes
0answers
29 views

Showing where complex function is analytic and differentiable.

I've been asked to show where the following function is analytic and differentiable; $$f(z) = x^4 + i(1-y)^4$$ for $z = x + iy$ First, I noted that $u(x,y) = x^4$ and $v(x,y) = (1-y)^4$. Then, I ...
1
vote
1answer
59 views

Complex Analysis - Definition of Singular Point

I have been reviewing Dennis Zill's Complex Analysis text and he defines a singular point as a point $z$ at which a function $f$ fails to be analytic. Now he goes on to talk about isolated ...
0
votes
1answer
34 views

One-sided total derivative

Given a function from half space into euclidean space: $f:\mathbb{H}^m\to\mathbb{R}^n$ Suppose its one-sided limit exists at a specific point: $\lim_{\mathbb{H}^m\owns v\to 0}\frac{1}{\lVert ...
0
votes
4answers
40 views

Induction of logarithmic derivatives of complex functions?

I am trying to use induction to prove the logarithmic derivative of a complex function (called $P(Z)$ here). I define a function $P(z) = (z-z_1)(z-z_2)...(z-z_n)$ and then I want to use induction on ...
0
votes
1answer
26 views

Finding the derivative of analytic polynomials

I have just started studying complex analysis and i am stuck with one question. My book says, the derivative of an analytic polynomial with respect to $z$ is equal to the partial derivative of that ...
0
votes
3answers
57 views

complex analysis: If $f$ is analytic and $\operatorname{Re}f(z) = \operatorname{Re}f(z+1)$ then $Im\;f(z) - Im\;f(z+1)$ is a constant

I am having trouble deciphering the reason behind a line in a complex analysis textbook (Complex made Simple by Ullrich, page 360 5 lines down in Proof of Theorem B, for those who are interested). ...
0
votes
0answers
12 views

Checking where Differentiable and Analytic

Describe the set of the points in the complex plane where the following functions of complex variable $z$ are differentiable and the sets of points where the functions are analytic. a) $f(z) = ...
1
vote
1answer
64 views

Derivative of $\operatorname{Log}(\operatorname{Log}(z^2))$

Please help me with this question: (i don't know how to start) Suppose that $f(z)$ = $\operatorname{Log}(\operatorname{Log}(z^2))$. Find $f'(z)$ where it exists, and determine the set of points at ...
1
vote
1answer
286 views

Complex integration around a branch point

I am confused about the "deformation" of a closed contour that my book is doing. For reference, it is example 2.4.3 on pg. 75-76 from this free online book. The example is the integration of 1/z ...
7
votes
2answers
303 views

Is there an analytic function satisfying $\,\,f\big(\!\frac 1 n\!\big)=\frac 1 {\sqrt{n}},\, \,n\in\mathbb N$?

Is there a function that is analytic in an open neighbourhood of $z=0$ and satisfies $$f\left(\!\dfrac 1 n\!\right)=\dfrac 1 {\sqrt{n}},$$ for all natural number $n$? I guess this problem requires ...
2
votes
0answers
84 views

Is it possible to switch limit from inside to outside of integral in this case?

Let $C$ be an open connected subset of $\mathbb{C}$. Let $f:[a,b]\times C \rightarrow \mathbb{C}$ be a function. Assume $f(-,z):[a,b]\rightarrow \mathbb{C}$ is continuous and $f(t,-):C\rightarrow ...
0
votes
1answer
22 views

Differentiating $f(z)=az^2+b\bar zz+c\bar z^2$

Suppose $f(z)=az^2+b\bar zz+c\bar z^2,$ where $a,b,c \in \mathbb C$ are fixed. By differentiating $f(z)$, show that f is complex differentiable at $z$ if and only if $bz+2c\bar z=0.$ So far I've ...
0
votes
1answer
55 views

Cauchy Riemann and Differentiability

Consider the following proposition. Proposition Let the function $$ f( z ) := u(x,y) + iv(x,y) $$ where $ z = x+iy $ be defined throughout in a $ \eta $-neighbourhood of $ c = a + ib $. Suppose ...
28
votes
7answers
8k views

What function can be differentiated twice, but not 3 times?

In complex analysis class professor said that in complex analysis if a function is differentiable once, it can be differentiated infinite number of times. In real analysis there are cases where a ...
0
votes
1answer
55 views

check if complex function is differentiable

The question is to check where the following complex function is differentiable. $$w=z \left| z\right|$$ $$w=\sqrt{x^2+y^2} (x+i y)$$ $$u = x\sqrt{x^2+y^2}$$ $$v = y\sqrt{x^2+y^2}$$ Using the ...
2
votes
1answer
35 views

$g(z) = \int_{0}^{2\pi}f(e^{i\theta})\frac{e^{-i\theta}}{e^{-i\theta}-\bar{z}}d\theta$ is antiholomorphic

I 've encountered this fact: if $z \in D(0,1) $ and $f$ is continous on $\partial D(0,1) $ then $$g(z) = \int_{0}^{2\pi}f(e^{i\theta})\frac{e^{-i\theta}}{e^{-i\theta}-\bar{z}}d\theta$$ is ...
2
votes
1answer
126 views

Differentiability vs Analyticity

What makes the crucial difference between the reals and the complex numbers is that the complex numbers are algebraically closed. So while going through all the proofs that "being holomorphic implies ...
0
votes
2answers
51 views

How to get the derivatives with respect to complex matrices

How could I get the derivative of the second term with respect to $\bar{\Delta}_k$ in the equation (19)? This result is obtained in the paper Robust Downlink Beamforming With Partial Channel State ...
1
vote
1answer
52 views

$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$

Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. I want to demonstrate that if $$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$$ then $f$ is linear. I know this is a well-known ...
1
vote
1answer
32 views

can the derivative of a closed complex contour at any point be zero?

If C is a closed contour in the complex plane parametrized by z(t)=u(t)+i*v(t), can there be any point where z'(t)=0?
1
vote
1answer
46 views

Taylor series expansion - application

I am working on the following: Let $f : \mathbb C \to \mathbb C$ be analytic. Suppose for all $z \in \mathbb C$ hold $f(2z) = 4f(z)$ and $f(1) = 1$. Then $f(z) = z^2$ for all $z \in \mathbb C$. I ...
0
votes
2answers
32 views

Constructing an antiderivative of a function if the contour integral depends on initial and final point

I am working on the following problem: Let $D \subset \mathbb C$ be a domain, $f: D \to \mathbb C$ a continuous function and $\gamma : [\alpha, \beta] \to D$ a contour. Assume that $\int_\gamma f$ ...
2
votes
1answer
52 views

Inequality involving derivative of a complex function

Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$ Furthermore, there is equality if and only if $f$ is linear. Any ...
2
votes
1answer
440 views

Finding the taylor series of $f(z) = 1/(1+z^2)$.

I am working on the following exercise: Find the Taylor expansion of the function $f(z) = \frac{1}{1+z^2}$ about $z = 3i$. We had the Taylor Series Theorem in the lecture: Let $D \subset ...
1
vote
1answer
36 views

Derivative of exp with definition of differentiability

Prove with the definition of differentiability that $\exp(z)$ is differentiable in $\mathbb C$ and $(\exp(z))' = \exp(z)$ for all $z \in \mathbb C.$ I tried: \begin{align*} \frac{\exp(z+h) - ...
0
votes
1answer
46 views

Complex differentiability equivalent to linear approximation

Let $G \subset \mathbb C$ be an open set and $f: G \to \mathbb C$ a complex function on $G$. Prove that the function $f$ is complex differentiable at a point $z \in G$ if and only if there exists a ...
1
vote
1answer
81 views

How to prove that a complex power series is differentiable

I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of ...
18
votes
1answer
288 views

Derivative of the Meijer G-function with respect to one of its parameters

Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in ...
1
vote
2answers
76 views

Calculate the derivative of a complex norm

I'm stuck with a rather trivial looking question. How do you calculate the derivative of the norm of a complex number to it self? Like in $$ \frac{d|a|^2}{da} = ? $$ I think it would give rise to a ...
1
vote
1answer
15 views

How to show that $\frac{\partial}{\partial y}\left(\int_{0}^{y}\frac{1}{x+it-2}dt\right)=\frac{1}{x+iy-2}$

I'm trying to show that $$\frac{\partial}{\partial y}\left(\int_{0}^{y}\frac{1}{x+it-2}dt\right)=\frac{1}{x+iy-2}$$ In an area that doesn't contain the point $2+0i$. If the function under the integral ...
0
votes
1answer
49 views

Questions regarding complex differentiablity of complex functions with differentiable real/imaginary parts

I'm studying about Complex functions and I came across these two following questions which I haven't really been able to solve. Let $f\left(z\right)=u\left(x,y\right)+iv\left(x,y\right)$ be defined ...
3
votes
1answer
81 views

Proving a sufficient condition for complex differentiability

I'm trying to show that given $f=u+iv:\mathbb{C}\to\mathbb{C}$ and $z_{0}\in\mathbb{C}$ if $u,v$ are differentiable (as functions $\mathbb{R}^{2}\to\mathbb{R})$ at ...
1
vote
1answer
45 views

Complex derivative of a function

I've got the following question. We have a function $f(x,y) :R^2 \rightarrow R^2$ and we treat R^2 as complex numbers. We know that $\frac{df}{dy} = z_1$. Knowing that $f$ has a complex derivative, ...
0
votes
1answer
41 views

Can a complex function be complex-differentiable at a point and not in a neighborhood?

Is it possible for a function $f:\mathbb{C} \to \mathbb{C}$ to be complex-differentiable at a point $z_0\in \mathbb{C}$ without being analytic in a neighborhood of $z_0$? How can we prove this?
2
votes
2answers
43 views

How does this differentiation come about ?

The question is that: If $f(z)$ is analytic, show that $\frac{\partial f}{\partial \bar z} = 0$ Now, assuming $f(z) = u + iv$ $\frac{\partial f}{\partial \bar z} = \frac{\partial}{\partial \bar ...
0
votes
1answer
46 views

Complex Analysis

Determine the derivative of the following functions and state where they are analytic. $$f(z) = \log(z^3) \quad \Rightarrow f'(z) = \frac{3z^2}{z^3} = \frac{3}{z}$$ Hence, this function is analytic ...
2
votes
1answer
116 views

Determine all points in the complex plane where this function is differentiable.

With the function being: $$f(z) = \frac{1}{z^2-iz+1} + \sin(\cosh(z))$$ I'm a little unsure of how to even approach this. I'm guessing that when $z^2-iz+1=0$ the function is not differentiable. Also ...
0
votes
1answer
55 views

derivative of complex function

i want to ask question related to about derivative of complex function: if $f(z)$ is differentiable in a connected open set $R$ and if $f'(z)=0$, through $R$,then $f(z)$ is constant in $R$ i ...
3
votes
2answers
262 views

Complex differentiation under the integral sign (Ahlfors)

In Ahlfors' Complex Analysis text, page 202, he claims that in $\{ \Re z>0 \} $ $$\frac{d}{dz} \int_0^\infty \frac{2 \eta}{\eta^2+z^2} \frac{\mathrm d \eta}{e^{2 \pi \eta}-1}=- \int_0^\infty ...
1
vote
2answers
218 views

Computation of the Wirtinger derivative of a product

Let's have a function $f = (A/2)\phi\bar{\phi}$, where $\phi=\phi(z)$ is a complex-valued scalar field. I need to obtain $df/d\phi$. If I treat the real and imaginary parts of $\phi$ as independent ...
1
vote
1answer
110 views

Cauchy Derivative Estimates for entire functions with a bound.

The problem statement: Assume $f$ is an entire function and that there is an $n \in \mathbb{N}$ and a $C < \infty$ such that for $z \in C$ $$|f(z)| \le C ( 1+|z|^n)$$ Also assume that $f$ is never ...
0
votes
1answer
75 views

Is my application of Cauchy-Riemann right?

Question: Given $f(z) = 3z^2 + 9z^3 -z$. 1. Find $f^\prime(z)$ 2. Find $f(z)$ when $z = 3 + 2i$ 3. Use Cauchy-Riemann to find if $f(z)$ is differentiable at $3 + 2i$ My Attepmt: 1. $f^\prime(z) ...
2
votes
2answers
408 views

When a limit does not exist, can its derivative be found?

I am learning derivatives of complex numbers (functions, actually) and what a learned community member pointed to me was that there is a subtle difference between finding derivatives of real ...
0
votes
2answers
334 views

f(z) can be expressed as u(x,y) , iv(x,y). Am I doing this right?

So, $f(z) = z^2 + 4z - 6i$ and I need to express this as $u(x,y)$ , $iv(x,y)$. So, I plug in $z = x+iy$ and simplify. I am left with this: $f(x+iy) = x^2 - y^2 + 2xy + 4x + 4iy -6i$. Now, how do ...
1
vote
1answer
31 views

Calculate the $r$-derivative of the function $f$

Let $f$ be an analytic function defined by $$f(s)=g(s)∑_{n=1}^{∞}a_{n}/n^{s}$$ where $∑_{n=1}^{∞}a_{n}/n^{s}$ is an absolute convergent series for $Re(s)>1$. I have the following question: ...
2
votes
1answer
143 views

Wirtinger derivative of composition of functions

So I have a very basic question : let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a $C^1$ function, and let $g : \mathbb{C} \rightarrow \mathbb{R}$ be defined by $g(z)=h(z \overline{z})$. I want to ...
11
votes
2answers
281 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 $f^3$ ...
1
vote
2answers
867 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
73 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 ...