# Questions tagged [cauchy-riemann-equations]

For questions on the solving/usage/applications of the Cauchy-Riemann equations. Use this tag alongside the complex analysis tag.

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### Previous year queation complex numbers [closed]

my attempt I solved the derivation part but unable to prove that the Ux=Vy AT f(0)...kindly direct me how to solve it..... I got Ux=(X^4 +3X^2Y^2 + 2XY^3)/(X^2 + Y^2)^2 Uy= /fract{-Y^4 -2X^3Y -3X^2Y^2}...
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### How to show a given function satisfies the Cauchy Riemann equation at a point, but is not differentiable at that point.

This is my question: Prove that the function $f(z) = u+iv$, where $$f(z) = \begin{cases} \frac{x^3(1+i) - y^3(1-i)}{x^2+y^2}, &\text{ if }z\neq 0,\\ 0 &\text{if }z=0.\end{cases}$$ satisfies ...
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### to check differentiability in complex analysis using c- r equations [closed]

Prove that the function f(z)=x^2*y^2 is differentiable at all point in each of the coordinate axis but not at any other points in the complex plane
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### Find an example of an entire function such that $f(z)=-f(iz)$ for all $z$

Is there a systematic way to find an example of an entire function $f$ such that $f(z)=-f(iz)$ for all $z$? By testing at monomials, we find that $f(z)=z^6$ is a solution. But how can we find other ...
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### How should I structure this complex analysis proof?

Suppose that $f$ is an entire function (i.e. analytic over the entire complex plane). Show that if $$f(z)=\overline{i f(-z)}$$ then $f$ must be a constant function of the form $$f(z)=c(1-i)$$ ...
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### If $f(z)$ is analytic in a domain D then show that $f^2(z)$ is analytic there.

If $f(z)$ is analytic in a domain D then show that $f^2(z)$ is analytic there. Let $f(z)=u+iv$, then $f^2=u^2-v^2+2iuv=X+iY$, say then by CR equation due to analyticitiy, it can be shown that CR ...
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### Holomorphism of an equation

I am confused about the property of holomorphism of complex functions. Take the function $\frac{1}{z}$ as an example. This function satisfies the Cauchy-Riemann equations and is thus said to be ...
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### analyticity of f(x+iy)=x^3+ax^2y+bxy^2+cy^3

please have a look at the question, Necessary condition for analyticity of $f(x+iy)=x^3+ax^2y+bxy^2+cy^3$ to solve the question i started from f(x+iy)=u(x,y)+iv(x,y) since it is analytic it will hold ...
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### Pointwise Cauchy-Riemann equations suffice?

Another question just now reminded me of something I realized a while ago I didn't know how to do: Say $V\subset\Bbb C$ is open, $f:V\to\Bbb C$, $f=u+iv$, and at every point of $V$ the partials of $u$...
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### find all functions f, for which g is analytic

let f(z)=u+iv be analytic in open disk $D=\{z\in C s.t |Z|<1\}$. assume $u\neq v$ for all (x,y) in D. Find all such f, for which the funstion $g(z)=u^2+iv^2$ in analytic in D. My answer is: g is ...
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### Cauchy Riemann equations, application

For $u: \Bbb C \to \Bbb R$ with $$u(x + iy) = 2x^3 - 6xy^2 + x^2 - y^2$$ find a function $v: \Bbb C \to \Bbb R$ s.t. $f = u + iv$ is holomorphic. I see that I have to verify the Cauchy Riemann ...
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### Two different forms of Cauchy-Riemann equation:

Let $f=\frac{1}{1+z^2}$. I am trying to determine whether f is differentiable. $f=\frac{1}{1+z^2}=\frac{1}{1+(x+iy)(x+iy)}=\frac{1}{1+x^2-y^2+2ixy}=\frac{1+x^2-y^2-2ixy}{(1+x^2-y^2)^2-4x^2y^2}$ ...
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### Let $z = x + iy$, then $f(x) = x^2 - y^2$ is this analytic?

I do not know if I did this fully correctly this is what I have: $$f(z) = (x + iy)^2 - y^2 = (x^2 + 2xyi - y^2) - y^2$$ so $u(x,y) = x^2 - 2y^2$, $v(x,y) = 2xy$ so by C-R \frac{\partial u}{\...
### Find an expression for $f(z)$ using Cauchy-Riemann Equations:
Suppose that $f: \mathbb{C} \to \mathbb{C}$ with $f(z)=u(x,y) + iv(x,y)$ is differentiable and that $3u(x,y) + v(x,y) = 7$, $\forall z=x+iy \in \mathbb{C}$. Find an expression for $f(z)$. I've been ...