Intereting Posts

Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a – b|$
Understanding vector projection
Equal angles formed by the tangent lines to an ellipse and the lines through the foci.
Write formula in MSO expressing that graph is grid (many variants)
last two digits of $14^{5532}$?
Prove that the only homomorphism between two cyclic groups with distinct prime orders is the trivial one
To determine whether the integral $\int_0^{\infty} \frac{\sin{(ax+b)}}{x^p} \,\mathrm dx$ converges for $p>0$
How Find $3x^3+4y^3=7,4x^4+3y^4=16$
Understanding an example from Hatcher – cellular homology
Explicit formula for conformal map from ellipse to unit disc (interior to interior)
Find all solutions to the following equation: $x^3=-8i$
Simplifying the integral $\int\frac{dx}{(3 + 2\sin x – \cos x)}$ by an easy approach
combining conditional probabilities
Prove inequality $\arccos \left( \frac{\sin 1-\sin x}{1-x} \right) \leq \sqrt{\frac{1+x+x^2}{3}}$
Are these two statements equivalent?

Dirichlet’s problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we know that the constructed function has uniform limit which is the given continuous function as $r\to 1$, and we can complexify the solution by taking the Schwarz integral formula. Now (in this very particular case) **does the imaginary part has a uniform limit**?

If it does, why don’t we prove the uniqueness of the solution by a simple application of the **Cauchy integral formula**? Here is the proof that I reckon to be reasonable(which is what I expose here to be criticised)

We construct the Schwarz integral, or

$\frac{1}{2\pi}\int _{0}^{2\pi}\frac{\exp(it)+z}{\exp{(it)}-z}u(\exp{(it)})\mathrm{d}t$.

This function is an analytic function in the interior unit disk, and is continuous in the closed unit disk(its closure). Let this analytic function denoted by $f$, and another analytic function that has the same uniform limit function on the circle be $v$. Then $u-v$ is an complex analytic function in the interior of the unit disk, continuous on the closure, with boundary value identically zero. An application of Cauchy’s integral formula gives rise to the conclusion that $u-v$ in identically zero on the entire disk.

Another question being if the boundary function has finite jump discontinuities, what is the **behaviour of the convolution with the Poisson kernel near the discontinuous points**?

- Proving that $ \int_{0}^{\pi/2} \frac{\mathrm{d}{x}}{\sqrt{a^{2} {\cos^{2}}(x) + b^{2} {\sin^{2}}(x)}} = \frac{\pi}{2 \cdot \text{AGM}(a,b)} $.
- Determining Laurent series $f(z)=\frac{1}{(z-2)(z-3)}$.
- If $f$ is holomorphic and $\,f'' = f$, then $f(z) = A \cosh z + B \sinh z$
- The Riemann zeta function $\zeta(s)$ has no zeros for $\Re(s)>1$
- Finding $f$ such that $ \int f = \sum f$
- What are the subsets of the unit circle that can be the points in which a power series is convergent?

- Evaluating the integral $\int_{-1}^{1} \frac{\sqrt{1-x^{2}}}{1+x^{2}} \, dx$ using a dumbbell-shaped contour
- Branch cut and $\log(z)$ derivative
- What is a simple form of this integral?
- Absolute convergence of Fourier series of a Hölder continuous function
- Residue of $z^2 e^{1/\sin z}$ at $z=\pi$
- Solving the complex polynomial
- How do I integrate $\int_{0}^{1}\!\sin x^2\,dx$?
- Pointwise but not uniform convergence of a Fourier series
- Fourier Transform of Schwartz Space
- Adjoint of multiplication by $z$ in a Hilbert Space (Bergman space)

No, the conjugate of a harmonic function that is continuous up to the boundary need not be continuous up to the boundary, or even bounded. This is related to the fact that the Hilbert transform does not preserve continuity (though it does preserve Hölder continuity of exponents $\alpha\in (0,1)$).

Here is an example. Let $F$ be a conformal map of the unit disk onto the domain bounded by the curves $y=0$ and $y=1/(1+x^2)$. Clearly, $F$ is not bounded: there are two points on the boundary of the unit disk which are sent into infinity by $F$. However, the imaginary part of $F$ is continuous: at the aforementioned points it approaches $0$. Thus, $\operatorname{Im}F$ is a harmonic function that is continuous in the closed unit disk, but whose conjugate is not bounded in the unit disk.

Another example is given by conformal map $G$ onto this rectangle with infinitely many slits:

Here the real part of $G$ is continuous up to the boundary, but the imaginary part is discontinuous, despite being bounded. (There is a point of unit circle where its cluster set is an interval of the size equal to the height of these slits.)

Typical example: consider the behavior of $\arg z$ in the upper halfplane as $z$ approaches $0$. Note that the boundary values of $\arg z$ have a jump discontinuity at $0$: they jump from $0$ to $\pi$. When $z$ approaches $0$ from within the domain, we can get any number between $0$ and $\pi$ as a limit. Or have no limit at all, if the curve of approach is zig-zagging.

This behavior is common. In fact, one can use $\arg z$ to prove this claim: subtracting an appropriate version of it from the boundary values removes the discontinuity, and then the behavior of function becomes clear: it consists of continuous part, plus a part that behaves like $\arg z$ above.

- Vandermonde identity corollary $\sum_{v=0}^{n}\frac{(2n)!}{(v!)^2(n-v)!^2}={2n \choose n}^2$
- When do weak and original topology coincide?
- On the growth of the Jacobi theta function
- Ambiguity of notation: $\sin(x)^2$
- Relative sizes of sets of integers and rationals revisited – how do I make sense of this?
- How to evaluate the following integral using hypergeometric function?
- Two problems about Structure Theorem for finitely generated modules over PIDs
- Irreducible polynomial over an algebraically closed field
- What does the function f: x ↦ y mean?
- Does the triangle inequality follow from the rest of the properties of a subfield-valued absolute value?
- Can it be determined that the sum of the diagonal entries, of matrix A, equals the sum of eigenvalues of A
- the number of loops on lattice?
- Generalization of “easy” 1-D proof of Brouwer fixed point theorem
- PDE Evans, 1st edition, Chapter 5, Problem 14
- Limit of $n-1$ measure of the boundary of a sphere