Intereting Posts

Normed vector space with a closed subspace
Find a positive integer $n$ such that $ϕ(n) = ϕ(n + 1) = ϕ(n + 2)$
Infinity as an element
Logical Form – Union of a Set containing the Power Set with Predicate/Propositional Function
Guide to mathematical physics?
Recursion relation for Euler numbers
Polar to cartesian form of $ r = \sin(2\theta)$
Is there an intuitionist (i.e., constructive) proof of the infinitude of primes?
What is the periodicity of the function $\sin(ax) \cos(bx)$ where $a$ and $b$ are rationals?
Pointwise convergent and total variation
Can a vector subspace have a unique complement in absence of choice?
Evaluating $\lim_{x \to 0}\frac{(1+x)^{1/x} – e}{x}$
how do we assume there is infinity?
Deriving cost function using MLE :Why use log function?
Finding the number of elements of order two in the symmetric group $S_4$

I’m trying to learn a bit of complex analysis, and this idea has got me stuck.

I would like to show that, for $u$ a function of a complex variable $z$, that $u(z)$ and $u(\bar{z})$ are simultaneously harmonic.

I try writing $u(z)=a(z)+ib(z)$. Assuming $u(z)$ is harmonic,

$$

\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0.

$$

Also, I think

$$

\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 a}{\partial x^2}+i\frac{\partial^2 b}{\partial x^2},

\qquad

\frac{\partial^2 u}{\partial y^2}=\frac{\partial^2 a}{\partial y^2}+i\frac{\partial^2 b}{\partial y^2}.

$$

I don’t understand how to use this to show $u(z)$ and $u(\bar{z})$ are simultaneously harmonic. Aren’t these the same function $u$? Shouldn’t that be independent of whether you plug in $z$ or $\bar{z}$? Thanks.

- Singular points
- How to choose a proper contour for a contour integral?
- Which meromorphic functions are logarithmic derivatives of other meromorphic functions?
- $f, g$ entire functions with $f^2 + g^2 \equiv 1 \implies \exists h $ entire with $f(z) = \cos(h(z))$ and $g(z) = \sin(h(z))$
- Simplification of $G_{2,4}^{4,2}\left(\frac18,\frac12\middle|\begin{array}{c}\frac12,\frac12\\0,0,\frac12,\frac12\\\end{array}\right)$
- Is every entire function is a sum of an entire function bounded on every horizontal strip and an entire function bounded on every vertical strip?

- Difference between $\mathbb C$ and $\mathbb R^2$
- Prove that $\zeta(4)=\pi^4/90$
- Evaluating the series $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $
- integral to infinity + imaginary constant
- How do I find the series expansion of the meromorphic function $\frac{1}{e^z+1}$?
- Uniform convergence of Taylor series
- Determine the Winding Numbers of the Chinese Unicom Symbol
- Riemann Zeta Function and Analytic Continuation
- Proving $f$ has at least one zero inside unit disk
- how to find a complex integral when the singular point is on the given curve

The Cauchy-Riemann equations are sufficient to show that $u$ is harmonic. Define $v(x,y)=u(\bar{z})$.

Now $z\ne \bar{z}$ so there’s no reason to expect $u(z)=u(\bar{z})$ (take $u(z)=z$ and $z=i$ for instance), so we are not looking at the same function with $v(x,y)$. Note that $(\partial/\partial x)^2 v=v_{xx}=u_{xx}$ but using the chain rule gives:

$$v_{yy}=\frac{\partial}{\partial y}\left(-u_y\right(\bar{z}))=(-1)^{2}u_{xx}(\bar{z})$$

whence the harmonicity of $u(z)$ entails that of $u(\bar{z})$. (Keep in mind I abused the distinction between $\mathbb{C}$ and $\mathbb{R}^2$ quite severely here.)

Just use the chain rule. Note first that $u(\overline{z}) = u\circ\overline{\cdot} = u(x,-y)$. The $x$ derivative of $u\circ\overline{\cdot}$ is the same as the $x$ derivative of $u$ and the $y$ derivative picks up a factor of $-1$ has $(u\circ\overline{\cdot})_y = -u_y$. The second $y$ derivative picks up another $-1$, which gives $(u\circ\overline{\cdot})_{xx} + (-1)^2(u\circ\overline{\cdot})_{yy} = u_{xx} + u_{yy} = 0.$

- Differential operator applied to convolution
- Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?
- Is there a lower-bound version of the triangle inequality for more than two terms?
- How to solve this inequality.
- T$\mathbb{S}^{n} \times \mathbb{R}$ is diffeomorphic to $\mathbb{S}^{n}\times \mathbb{R}^{n+1}$
- Explicitly finding the sum of $\arctan(1/(n^2+n+1))$
- How to prove that a set R\Z is open
- Intuitive Explanation why the Fundamental Theorem of Algebra fails for infinite sums
- “Counting Tricks”: using combination to derive a general formula for $1^2 + 2^2 + \cdots + n^2$
- Homotopy groups of $S^2$
- Why does the semigroup commute with integration?
- Probability that the first digit of $2^{n}$ is 1
- Basis for dual in infinite dimensional vector space.
- Solid body rotation around 2-axes
- Average distance between two random points in a square