# Tagged Questions

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### 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 $f'(z)=u_x+iv_x$.
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### 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 ...
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### 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
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### 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 ...
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### 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 ...
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### 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: ...
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### 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, ...
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### 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 ...
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### 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:
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) - ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...
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 ...