0
votes
0answers
26 views

proof of derivative of a complex function

suppose $u(x,y)$ is harmonic in a domain $D$ and $v(x,y)$ is an harmonic conjugate of $u$. Let $f(z)=u(x,y)+iv(x,y)$. Prove $f'(z)=u_x+iv_x$.
1
vote
1answer
33 views

Differentiability of non-analytic complex functions

Any complex function that is analytic on an open set is differentiable on that set. But can a function fail to be analytic on an open set but still be differentiable? For example, the function ...
1
vote
1answer
19 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
1
vote
2answers
42 views

Using complex derivative to shows that a function is constant

If we know $\frac{\partial f}{\partial z} \equiv f'(z)=0$ where $f(z)=u(x,y)+iv(x,y)$ why do we need to check the Cauchy Riemann equations are all equal to zero, before concluding that $f$ is ...
1
vote
0answers
32 views

Zeros of the derivatives of a finite Blaschke product.

Let $B$ be an $n$ degree finite Blaschke product. By considering the level curves of $B$, one can show that $B'$ has $n-1$ critical points in the disk (counting multiplicity). Is anything known ...
2
votes
0answers
35 views

Proof of Cauchy-Riemann equations using differentials as quotients?

In my analysis 2 book the proof goes like this: If a complex function $f = P(x,y) + iQ(x,y)$ is differentiable at a point $z$, then $$ \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} ...
2
votes
2answers
71 views

A function is real-differentiable iff it has a complex-differentiable extension

Is this conjecture true? A function $f:\Bbb R\to\Bbb R$ is real differentiable at $a$ if and only if there exists a complex-differentiable function $g:A\to\Bbb C$ for some neighborhood of $a\in ...
-1
votes
1answer
36 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...
0
votes
2answers
44 views

If the real part of analytic function satisfies $u_x=u_y$, then the function is linear

Let f(z) be analytic function and $\forall z=x+iy\in\mathbb C, u_x=u_y$ ($u_x=\frac{\partial f}{\partial x},u_y=\frac{\partial f}{\partial y}$. Prove that $f(z)=az+b$. I thought using Cauchy ...
1
vote
1answer
48 views

Proving that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to z_0}\frac{f^\prime(z)}{g^\prime(z)}$

Let $f,g$ both analythic in neighbourhood of $z_0$ and they both have zero of multiplicity $n$ in $z_0$. Prove that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to ...
0
votes
0answers
45 views

Cauchy–Riemann equations

What are the steps to find many functions that satisfy Cauchy–Riemann equations at a point $$z=z_0$$ but are not differentiable at that point
0
votes
1answer
24 views

Check complex differentiability

I am trying to take a derivative w.r.t $z\in\mathbb{C}$ of the following map: $z\mapsto \sum_{j=0}^{\infty}\lambda_{j} (T(\psi+zh))_{j}$ where $(\lambda_{j})$ is a bounded sequence, $T$ is a ...
1
vote
2answers
46 views

Proving that a function is nowhere differentiable

I'm just going over a bit of revision for an upcoming exam, and I just wanted to verify whether my working/argument was sound. I've been asked to show that $f(z) = \frac{1}{\overline{z}}$ is nowhere ...
0
votes
2answers
61 views

Derivative of complex number

Can anyone please help obtain the partial derivative of $\Im(\arctan(x+iy))$ with respect to $x$ and $y$, respectively. actually this value can be plotted in the $x$-$y$ plane, as shown by follows: ...
1
vote
1answer
65 views

Residue of this function for $z_0=0$

I have this function $$\frac{\sin (2z)-2z}{(1-\cos z)^2}$$ I want to find its residue around $z_0=0$, however I've been battling it for hours but I get nowhere. I've tried finding its Laurent series, ...
0
votes
1answer
40 views

Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function

In Walter Rudin's Principles of mathematical analysis Exercise 5.21, it is proved that for any closed subset $E\subseteq \mathbb{R}$, there exists a smooth function $f$ on $\mathbb{R}$ such that ...
1
vote
1answer
54 views

Limit differentiable?

What is a sufficient condition for the limit to be differentiable? Certainly pointwise convergence of derivatives is not enough: Also uniform convergence is not enough:
0
votes
1answer
28 views

Showing the Clairaut theorem in higher dimensions — partials commute

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
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
36 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
107 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
2answers
72 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
52 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
34 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 ...
1
vote
3answers
75 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
18 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
68 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
428 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
361 views

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

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 numbers $n$? I guess this problem ...
2
votes
0answers
87 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
27 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
62 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 ...
30
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
72 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
39 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
148 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
62 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
56 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
36 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
52 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
34 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
54 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
908 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
41 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
57 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
114 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 ...
19
votes
1answer
398 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
122 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 ...
2
votes
1answer
17 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
54 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 ...