In Complex Analysis by Kodaira, a more powerful version of Cauchy’s Integral Theorem (and consequently formula) was proven. The result generalizes the theorem to the boundary of an open set as follows Let $D$ be a domain and $\overline{D}$ be it’s closure. Suppose that $f:\overline{D} \rightarrow \mathbb{C}$ is holomorphic in $D$ and continuous on $\overline{D}$. […]

I was going through my introductory calculus book(for high-school student) by a Russian author(N.Psikunov) where I encountered a theorem without proof named: Weierstrass Approximation Theorem So how can we apply this theorem and apply it to piece wise functions?(any general approach?)(say a simple function like |x|)

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented. The numerator is not analytic in $\Gamma$ so we can’t use Cauchy integral formula. I’m thinking maybe I shold use residue theorem. But then I have these two questions: Should I look for the Laurent series of $e^\frac{1}{z}$ around $z=0$? What if I look […]

Find all entire functions with $f(z) = f(\frac{1}{z})$, for all $z\ne 0$. I tried to use the power series of $f$ but this did not help. I also tried to use Liouville’s Theorem on $\frac{f(z)}{f(\frac{1}{z})}$ but then I have to deal with possible singularities. Can someone help me with this problem?

Let $K$ be a compact subset of $\Bbb C$. Let $\mathcal P(K):=$ closure in $||\cdot||_{\infty}$ of all complex polynomials on $K$. Note that $\mathcal P(K) = C(K)$ do not generally hold since Stone-Weierstrass do not apply to complex polynomial. What is the character space $\Phi_{\mathcal P(K)}$ of $\mathcal P(K)$? It is commonly known that $\Phi_{C(K)}=\{ […]

Let $f$ be an analytic function on the open unit disc and let $g$ be an analytic function on the complement of its closure. Further assume that the two functions have a the same continuous limit on the common boundary of their domain. Is it possible to build an entire function that settles with $f$ […]

Evaluate $$\int \cos^2\theta \space d\theta$$using complex numbers. My attempt: $\displaystyle\int \cos^2\theta\space d\theta=\displaystyle\int\left(\dfrac{e^{i\theta}+e^{-i\theta}}{2}\right)^2\space d\theta$ Then I tried to simplify the integrand: $\displaystyle\int\dfrac{1}{4}(e^{i\theta}+e^{-i\theta})^2\space d\theta \\=\dfrac{1}{4}\displaystyle\int(e^{2i\theta}+2e^{i\theta}\cdot e^{-i\theta}+e^{-2i\theta}\space d\theta) \\=\dfrac{1}{4}\displaystyle\int(e^{2i\theta}+e^{-2i\theta}+2)\space d\theta$ At this point, I’m not sure how to proceed though. I haven’t learned how to integrate terms with $i$ in them yet.

Solve: $$ z^3 – 3z^2 + 6z – 4 = 0$$ How do I solve this? Can I do it by basically letting $ z = x + iy$ such that $ i = \sqrt{-1}$ and $ x, y \in \mathbf R $ and then substitute that into the equation and get a crazy long […]

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula $$\sum_{k = 1}^\infty \frac{\sin(kz)}{k} = \frac{\pi – z}{2}$$ using complex analysis for some $z$. In other words, it can be shown […]

I noticed someone do this from one of the questions is asked on here i had: $$e^z = -0.5$$ $$e^z = 0.5e^{i\pi}$$ which magically became: $$z = \ln\left(\frac12\right) + iπ + 2ikπ$$ does this mean that if i have: $$e^z = -r = re^{i\pi} = \ln(r) + iπ + 2ikπ$$ Thanks for any help you […]

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