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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.

In the field of complex analysis in mathematics, the Cauchy–Riemann equations (or, C-R equations) consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. Cauchy-Riemann Equations are discovered by the French mathematician Augustin Louis Cauchy $(1789-1857)$ and the German mathematician Georg Friedrich Bernhard Riemann $(1826-1866)$.

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  • Suppose that $$f(z)=f(x+iy)=u(x,y)+i~v(x,y),$$is differentiable at the point $~z_0=x_0+i~y_0)~$. Then the partial derivatives of $~u,~\text{and}~v~~$ exist at the point $~(x_0,y_0)~$, and can be used to calculate the derivative at $~(x_0,y_0)~$. That is, $$f'(z_0)=u_x(x_0,y_0)+~i~v_x(x_0,y_0)\qquad .. .....(1)$$ and also $$f'(z_0)=u_y(x_0,y_0)+~i~v_y(x_0,y_0)\qquad..........(2)$$ Equating the real and imaginary parts of Equations $(1)$ and $(2)$ gives the so-called Cauchy-Riemann Equations: $$u_x(x_0,y_0)=v_y(x_0,y_0)\qquad \text{and} \qquad u_y(x_0,y_0)=-v_x(x_0,y_0)$$

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C-R equation has a lots of application in Analysis, Fluid Dynamics and many other fields of Mathematics and physics. For more details please find the last two references.

References:

https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations

http://www.ijmetmr.com/oloctober2015/NaisanKhalafMosah-BShankar-A-30.pdf

https://www.quora.com/What-are-the-applications-of-the-Cauchy-Riemann-equations

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Is there any connection between Green's Theorem and the Cauchy-Riemann equations?

Green's Theorem has the form: $$\oint P(x,y)dx = - \iint \frac{\partial P}{\partial x}dxdy , \oint Q(x,y)dy = \iint \frac{\partial Q}{\partial y}dxdy $$ The Cauchy-Riemann equations have the following form:(Assuming $z = P(x,y) +…
Jichao
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Why isn't $f(z)=\bar{z}$ complex differentiable

It is quite easy to see why $f(z)=\bar{z}$ isn't complex differentiable. $\frac{\partial{u}}{\partial{x}}=1\neq-1=\frac{\partial{v}}{\partial{y}}$ But I struggle to see why this is the case. Visualization of the conjugate function in my head is…
Brian010515
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Prove that $f(z) = |z|^2$ is differentiable only at the origin

Question: I need to show that $f(z) = |z|^2$ is differentiable only at the origin Sufficient condition for differentiability: The sufficient condition for a function $f(z) = u + iv$ to be analytic at all the points in region $R$ are $\displaystyle…
Utshaw
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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 $u$ and $v$ exist and satisfy the Cauchy-Riemann…
David C. Ullrich
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Cauchy Riemann equations necessary and sufficient condition?

I was taught that $f(z)$ is differentiable at $z_0=x_0+y_0$ iff Cauchy Riemann equations hold at $(x_0,y_0)$. However, I was shown this example: $f(z)=\frac{\operatorname{Re}(z) \cdot \operatorname{Im}(z)}{z}, z \neq 0$ and $f(0)=0$. So…
Hjersh
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Coordinates Free Cauchy-Riemann equations

Cauchy-Riemann equations can be seen as a system of 2 PDEs for two functions on the plane: $$ \begin{align*} L_{\frac{\partial}{\partial x}}(u) &= L_{\frac{\partial}{\partial y}}(v) \\ L_{\frac{\partial}{\partial y}}(u) &=…
Athena
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Why does Cauchy-Goursat fail in real case?

I was reading about Cauchy-Goursat Theorem proof (the version that works with triangles or rectangles) and I couldn't determine when it fails when we want to replicate it in the real world. In this proof, we state: $$ f(z) = f(z_0) + f'(z_0)(z -…
Johhny Mc
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Holomorphic function preserving real and imaginary axis

Let $f$ be a holomorphic function such that $f(\mathbb{R})\subseteq \mathbb{R}$ and $f(i\mathbb{R})\subseteq i\mathbb{R}$. Prove that $f(-z)=-f(z)$. We want to prove $g(z)=f(-z)+f(z)$ is zero. Notice that $g(0)=2f(0)=0$, because $f(0)\in…
Kadmos
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Prove an entire function is constant on complex plane

Let $f(z)$, $F(z)$ be two analytic functions on $\Bbb C$ satisfies $f(z)=F(\overline{f(z)})$. Here $\overline{f(z)}$ is the complex conjugate of $f(z)$. Prove that $f(z)$ is constant on $\Bbb C$. I tried to use Cauchy-Riemann equations but I didn't…
QIRUN CONG
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generalization of Cauchy-Riemann conditions

Esteemed experts, Please excuse the ignorance and language of a poor physicist. As we know, the real $u(x,y)$ and imaginary $v(x,y)$ parts of an analytic function (in some domain) satisfy the Cauchy-Riemann (CR) conditions $\frac{\partial…
user43680
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Why is $\int_\Gamma \frac{\log(b-az)}{z} dz=2\pi i\log(b)$?

Let $a\leq b$ and $\Gamma$ be the unit circle (in the complex plane). I found that $\int_\Gamma \frac{\log(b-az)}{z} dz=2\pi i\log(b)$. It seems like Cauchy's integral formula has been used here. I want to know why the conditions of Cauchy's…
Mimimi
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What does inequality of complex numbers mean?

This problem is taken off the 2008 problem set for Math 185 in the University of Chicago (PDF link via uchicago.edu) (problem 4 on page 3): Suppose $f$ is analytic on a domain $\Omega \subset \mathbb{C}$ containing the unit square of the complex…
Cognoscenti
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Deriving the Cauchy-Riemann Equations

I have learned that one can derive the Cauchy-Riemann equations by using the "typical" definition of differentiation (i.e. $f'(z)=\lim_{h \rightarrow 0} \frac{f(z+h)-f(z)}{h}$) by letting $h=h_1+i0$ and calculating the limit and then letting…
user1493768
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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 exist and the Cauchy-Riemann rules must be…
user1107523
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2 answers

Wirtinger derivative form of Cauchy–Riemann equations

I'm trying to understand the Cauchy-Riemann equations using the traditional $u, v$ form and the Wirtinger derivative form. Taking $\ln|z|$ as an example function, for the normal $u, v$ form I have: $$\begin{align}u(x,y) &= \ln|x + iy|\\ v(x,y) &=…
Jay Lemmon
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