# 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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### Let $f, g: D \rightarrow \mathbb{C}$ be holomorphic functions on the domain $D \subseteq \mathbb{C}$ such that $f'(z) = g'(z)$ for each $z \in D$

(a) Given is the function $f: \mathbb{C} \rightarrow \mathbb{C}$ with the formula $f(z) = \frac{1 - i}{4} \left( z^2 + \bar{z}^2 \right) + \frac{1 + i}{2} |z|^2.$ Is $f$ an entire function? If not, ...
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### $f(x + iy) = u(x,y) + iv(x,y)$ holomorphic implies that $F(x,y) = (u(x,y), v(x,y))$ is differentiable

My question comes from section 1.6 of Complex Analysis (4th edition) by Serge Lang: Let $U$ be an open subset of $\mathbb{C}$ and let \begin{align*} f(x + iy) = u(x,y) + i v(x,y), \qquad x + iy \...
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### Derivative notation in section 1.2 of Stein-Shakarchi's Complex Analysis

I'm reading about holomorphic functions in section 1.2 of Complex Analysis by Stein and Shakarchi, and I am pretty confused about the derivative notation that the authors employ. In this section the ...
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### To find the constants of a function which is analytic

Find the real constants $a,b,c,d$ so that the function is analytic. $f(z)=x^2+axy+by^2+ i(cx^2+dxy+y^2)$ I know that,since the given function is analytic,we can use cauchy reimann equations to solve ...
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### How do I show that the Cauchy-Riemann Equations hold for any polynomial?

I'm self-studying complex analysis and trying to show that the Cauchy-Riemann Equations hold for any complex polynomial $$f(z) = a_n z^n + \dots + a_1 z + a_0$$ but I'm unsure how to actually get a ...
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### Can a) and b) real components of an complex equation? [closed]

I have a problem somewhere ... I missed a point.. So given are following equations: (with $C \to R$) (i) $a(z) = x^2 - y^2$ (ii) $b(z) = x^2 + y^2$ And I am supposed to find out if those ...
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How would one prove that Cauchy-Riemann equations hold for any holomorphic functions? I used the definition of complex differentiability to achieve the second equation ( $\frac{\partial u}{\partial y}=... • 90 0 votes 1 answer 40 views ###$ f(z) = z^2 $show that image curves of lines$x=a$and$y=b$are perpendicular to each other. Little sidenote: Im new to function theory, so I'm inexperienced and don't have many tools regarding this topic.$\mathbb{C^{*}} =${$z \in \mathbb{C} | z \neq 0$} with the function$ f(z) = z^2 $. ... 2 votes 2 answers 147 views ### A set of "Cauchy-Riemann" conditions for quaternion function? I was trying to get a "Cauchy-Riemann" conditions for a quaternion function$f:\Bbb H\to\Bbb H$,$f(q)=m(q)+n(q)i+o(q)j+p(q)k$and$m, n, o, p:\Bbb H\to\Bbb R$. Defining the quaternion ... • 495 0 votes 2 answers 99 views ### Question about the C-R equation and the Analytic functions I am currently reading Complex Analysis by Stein and found the follow theorem (Theorem 2.4 on Page 13): Suppose that$f = u + iv$is a complex-valued function defined on an open set$\Omega$. If$u$... • 143 3 votes 2 answers 182 views ### Which holomorphic functions have constant argument on rays from the origin? On circles centered at the origin? Multiples $$f(z) = c z, \qquad c \in \Bbb C \setminus \{0\},$$ of the identity function on$\Bbb C \setminus \{0\}$trivially all satisfy the following special condition: Condition A: All points on a ... • 102k 0 votes 1 answer 50 views ### Function / question about Cauchy-Riemann equations $$f(x+iy) = \sin^2(x+y) + i\cos^2(x+y)$$ I have calculated $$\frac{\partial u} {\partial x} = 2\sin(x+y)\cos(x+y)$$ $$\frac{\partial u} {\partial y} = 2\sin(x+y)\cos(x+y)$$ $$\frac{\partial v} {\... • 357 1 vote 0 answers 44 views ### Are pointwise solutions also weak? Suppose the partials of u, v :\Bbb R^2\to \Bbb R exist and satisfy the Cauchy-Riemann equations everywhere, with u and v locally integrable. Is it true that u and v are also weak solutions? ... • 7,348 0 votes 1 answer 110 views ### How do you get the real part of function f(z) given the imaginary I have a function f(z) with imaginary part v(x,y) = \frac x{x^2 + y^2}. How do I find the real part of this function? I am trying to solve this using the Cauchy-Riemann equations. I have found ... • 29 2 votes 2 answers 656 views ### Question about f(z)=\exp (-\frac{1}{z^4}) Let f(z)=\exp (-\frac{1}{z^4}) for z\neq 0 and f(0)=0. I know this is a famous example and got asked a lot However my question is not about the origin, but what is the best way to actually show ... • 537 2 votes 0 answers 52 views ### Where is f(z) = z^k \Re(z) complex differentiable? [closed] I'm trying to figure out where f(z) = z^k \Re(z) is complex differentiable. I've tried to express z as z=x+yi, but I can't figure out how to apply the Cauchy-Riemann equations. • 39 0 votes 0 answers 78 views ### Cauchy-Riemann: total or partial derivative with respect to \bar z The Cauchy-Riemann conditions for a function f are respected if:$$\frac{\partial f}{\partial \bar z} = 0$$But is it a partial or a total derivative actually ? If it is partial, then I can safely ... 2 votes 1 answer 133 views ### Show that f(z) = \frac{\sinh (\sqrt z)}{\sqrt z} is holomorphic on \mathbb C I have to show that f(z) = \frac{\sinh (\sqrt z)}{\sqrt z} is holomorphic on \mathbb C. My attempt is the following: Let f(z) = \frac{g(z)}{h(z)}, where g(z) = \sinh (\sqrt z) and h(z) = \... • 486 0 votes 1 answer 57 views ### If f=u+iv and u and v satify the Cauchy-Riemann equations does that imply that f is analytic? or do we need more conditions [closed] If f=u+iv and u and v satisfy the Cauchy-Riemann equations does that imply that f is analytic? or do we need more conditions 0 votes 1 answer 76 views ### I need help evaluating the given function I was given the following problem and told to evaluate where C is the circle |z|=2 :$$ \int_{|z|=2} \frac{1}{z^2-1} dz$$I've tried solving using Cauchy's Integral Formula and I got the answers$$... 0 votes 1 answer 44 views ### Components of a 2D vector field satisfy Cauchy Riemann conditions implies the finite transformation is holomorphic I am reading M. Schottenloher's book on Conformal field theory. https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/01_978-3-540-68625-5_Ch01_23-08-08.pdf On page 19 after Proposition 1.12, ... 4 votes 3 answers 103 views ### Prove Cauchy-Riemann respected with$\frac{\partial f}{\partial \bar z} = 0$I struggle a lot with complex analysis currently. I need to find the domain of analycity of$f(z) =: u(x, y) + iv(x, y)$, so the function needs to be continuous at$z$, its partial derivates must ... 2 votes 1 answer 48 views ### Finding all functions making$u + iu$holomorphic I wish to find all possible$C^{1}$functions$u: \mathbb{R}^2 \rightarrow \mathbb{R} $s.t.$f(x + iy) = u(x,y) + iu(x,y)$is analytic/holomorphic/complex differentiable. This would occur if and only ... -1 votes 2 answers 101 views ### Why does path-independence of limits hold in the complex numbers but not in$\mathbb{R}^2\$?

In my studies of complex differentiation, I've come across the following paradox concerning real partial derivatives: In proofs of the Cauchy-Riemann equations, the fact that the limit \begin{align*} \...