I am learning complex analysis and currently reading Ahlfors’ Complex Analysis. I feel like I need to do more exercises than the five problems provided in each section. Therefore, I am seeking recommendations for complex analysis books with more exercises than Ahlfors’, but still at the same leve...

Let $X = (X_n)_{n\in\mathbb{N}} \sim \text{MC}(\lambda,P)$ be a Markov chain on the natural numbers such that $\forall i\neq 0,\, (p_{i,i+1}=2/3$ and $p_{i,i-1}=1/3)$ and $p_{0,1}=1$ (every other value of $P$ is zero). That is, a Makrov chain in $\mathbb{N}$ where the probability of moving away f...

Triangle $(\triangle POB)$ under a semi circle. $OP$ and $BP$ are extended to $OA$ and $BQ$. $AP = 5$ and $PQ = 7$. What is the area of the triangle It's a problem I stumbled upon on chance. I haven't been able to solve it since I am a little weak in geometry. The best approach I have tak...

In a derivation I am working on, I have encountered an integral of the form \begin{equation} \int_{0}^{\infty}e^{-a x^2-b\ \textrm{cosh}(x)}\ dx \end{equation} with $a$ and $b$ real and positive. I am not sure if there is a closed form. Any ideas on how to attack it?

I am trying to come up with a formula for the number of ways to create $k$ non-empty subsets from $n$ elements which are not necessarily distinct. For instance, let us say the numbers are $a_{1},a_{2},\ldots, a_{n}$, and the subsets are $l_{1},l_{2},\ldots,l_{k}$. Obviously the order of elements ...

Let $A \subseteq \mathbb{R}^n$ with $\text{reach}(A) > 0$ (see https://en.wikipedia.org/wiki/Reach_(mathematics) ). Define for any $\epsilon>0$, the "removal of $\epsilon$-thick boundary", $$ A^{-\epsilon} = \{ x \in A \colon B(x,\epsilon) \subseteq A \}. $$ The question is: Does $A^{-\epsilon}$ ...

I am self studying proofwriting in preparation for an algebra class, and I though it would be a good idea to work through an introductory textbook. I ended up choosing Proofs by Jay Cummings due to all the endorsements, and after speeding through the first chapter and first dozen or so exercises,...

I'm trying to investigate the asymptotic property of the following integral: $$ \int_0^{\theta} \sin^nx dx, \quad n\to +\infty $$ where $\theta \in (0,\pi/2)$ is a constant. For the case $\theta = \pi/2$, it is the Wallis integral and we have $$ \int_0^{\pi/2} \sin^nx dx \sim \sqrt{\frac{...

I came up with this while messing around with the $\gcd$ and $\operatorname{lcm}$ functions in Desmos. $$I(a)=\int\limits_{0}^{\frac{\pi}{2}}\frac{\operatorname{lcm}(a\cos x,a\sin x)}{a^2}dx$$ The function inside the integral always has a slope of $0$ but is very discontinuous. At higher values o...

Consider a surface of revolution $S$ with constant positive curvature and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with conjugate points $p,q$ anchored on $\partial X^3$ where $\partial X^3=X^3-(0,1)^3$ for $\mathrm{dist}(p,q)=\sqrt{3}$. What is $\rho_{\mathrm{max}}=\mathrm{max} ...

THIS IS NOT A HOMEWORK QUESTION This question comes from a university entrance exam from around 1910. No solutions were ever published. My own solution is given here, but I get the feeling that a more efficient solution would be possible. Question: Find the exact value of $\cos{\frac{2\pi}{15}}+\...

I always thought it was valid to prove vacuously true statements using proof by contradiction but now am not so sure. For example, consider the vacuously true statement $\forall x \in \emptyset, x + 1 = 0$. To prove it by contradiction, I assume for the sake of contradiction that $\exists x \in \...

I have a question about the following problem [Finite Group Theory, Martin Isaacs, Chapter 5]: Let $G$ be simple and have an abelian Sylow 2-subgroup $P$ of order $2^{5}$. Deduce that $P$ is elementary abelian. I report my reasoning: From a theorem on the decomposition of finite abelian groups, I...

From topology I know the Urysohn lemma. My question is if there is some sort of Urysohn lemma for smooth manifolds. To be more precise: Let M be a smooth manifold and suppose $A$ and $B$ are two disjoint closed subsets of $M$. Then there is a smooth function $f$ on $M$ so that $0 \leq f(x) \leq 1...

Specifically, I would like to know whether: $$z^a z^b = z^{a+b}$$ $$(zw)^a = z^a w^a$$ $$(z^a)^b = z^{ab}$$ hold true for any and all $z,w \in \Bbb{C}$ and $a,b \in \Bbb{R}$? Also, I'd be interested to see any others that hold.

I am new here so please don't close the question (instead please tell me how to improve it). So I know that for a subset to be a subspace it has to satisfy the following properties: Closed under scalar multiplication. Closed under addition. I however do not know how to go about determining wh...