# Tagged Questions

19 views

### Understanding why $\int_\gamma {dz \over z - a} = k 2\pi i$ for $\gamma$ a closed curve not passing through $a$

The following is a paraphrased proof from Ahlfors. I bolded the part that is confusing me and asked a question about it at the bottom of this post. Hypothesis: Let $\gamma$ be a closed curve that ...
24 views

### Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve

Hypothesis: Suppose that $F(z)$ has $f(z)$ as a derivative. Suppose further that $F(z)$ is analytic. Now consider the complex line integral $$\tag{1} \int_\gamma f(z)\ dz$$ Question: Does this ...
23 views

### Verifying a condition for which $\int_\gamma p\ dx + q\ dy$ depends only on endpoints

Hypothesis: Suppose there exists a function $U(x,y)$ in $\Omega$ with partial derivatives $${\partial U \over \partial x} = p \quad \quad {\partial U \over \partial y} = q$$ Goal: Show that the ...
17 views

Goal: I'm trying to verify the calculation claimed by Ahlfors that $$\int_\gamma f(z)\ dz = \int_\gamma (u\ dx - v\ dy) + i \int_\gamma (u\ dy + v\ dx)$$ Attempt: $$\int_\gamma (u\ dx - v\ dy) + i ... 1answer 18 views ### Some questions about proof of Theorem 2.43 in Baby Rudin I will include the proof here and highlight the parts that are giving me trouble. Theorem \hspace{5 pt} Let P be a nonempty perfect set in \mathbb{R}^k. Then P is uncountable. Proof ... 0answers 21 views ### A simple proof with directional derivatives Suppose \nabla f (x) \cdot d < 0, prove that there exists \delta > 0 such that$$f(x + \tau d ) < f(x)$$for all \tau \in (0, \delta) My proof consists of only a few lines. ... 2answers 29 views ### If f is continuos on an interval, is it then uniformly continuous I have a function, f that is differentiable on (0,5), and I know it is continuous on (0,5). Is it also uniformly continuous on (0,5)? I now know that it is not. Can someone give me a proof ... 1answer 24 views ### If f is continuous on (0,5), is it uniformly continuous on same interval I have a function, f that is differentiable on (0,5), and I know it is continuous on (0,5). Is it also uniformly continuous on (0,5)? I believe it is. I now know that it is not. Can someone ... 1answer 15 views ### Holder's Inequality Proof Verification Wikipedia outlines a nice proof of Holder's Inequality in the link provided. The fifth sentence in the proof reads: Dividing fâ and g by \|fâ\|_p and \|g\|_q, respectively, we can assume ... 0answers 26 views ### Prove that \lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) dx = 1 for t>0. Suppose f \in \mathcal{R} on [0,A] for all A < \infty, and f(x) \rightarrow 1 as x \rightarrow + \infty. Prove that \lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) ... 1answer 110 views ### Proof concerning equivalent definition of supremum using limits I believe I have a proof to the following theorem: Let S be a subset of \mathbb{R} that is non-empty and bounded above. s \in \mathbb{R} is the supremum iff s is an upper bound of S and for ... 1answer 21 views ### Verify why this is not a metric d(x,y)=\|x-y\|_p ( \|x\|_p=p^{-h} if x=p^h\dfrac{m}{n}). d(x,y)=\|x-y\|_p p is prime and \|x\|_p=p^{-h} if x=p^h\dfrac{m}{n}, where m, n are coprimes with p. This is not a metric because if x=y=p^k\dfrac{m}{n}, then x-y=0=p^0\dfrac0n. ... 1answer 28 views ### v is Conjugate Harmonic to u \implies f = u + iv is Analytic (Proof Verification from Ahlfors) Hypothesis: Let u and v be two functions from \mathbb{R}^2 to \mathbb{R} s.t.$$ \Delta u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0 $$and$$ \Delta v = ...
63 views

In a lecture in Topology I had earlier this week, I was told (without proof) that not every topological space $(X,O)$ is metrizable, i.e, it is impossible to find some metric $d$ such that $O$ and ...
35 views

### Orthogonal representation of finite operator

I would like to know if my proof is correct. Statement: Let $T$ be a finite rank operator on a Hilbert space $\mathscr{H}$. Show that $\forall \, h \, \in \mathscr{H}, \, T(h)$ can be written as ...
80 views

### $f$ continuous nowhere implies $f$ not Riemann integrable

I am trying to prove the statement in the title, without using the theorem which says that the set of discontinuities of a Riemann integrable function must have measure $0$. I am aware that similar ...
44 views

### Isometry is not surjective

According to the definition I am using, an isometry is a mapping $f:X \rightarrow Y$ between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$: $$d_{Y}(f(a),f(b)) = d_{X}(a,b)$$ for all $a,b \in X$ I ...
62 views

### Possible book correction or am I missing something?

Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
36 views

### Showing that $\log \log(z)$ is Analytic (Proof Verification)

Goal: Convert $\log \log (z)$ into a single-valued function defined on a suitable region of $\mathbb{C}$ and then prove that it is analytic. Attempt: As has been demonstrated elsewhere, we have ...
49 views

### Showing that $\sqrt{1+z} + \sqrt{1-z}$ is Analytic (Proof Verification)

Ahlfors: Give a precise definition of a single-valued branch of the function $\sqrt{1+z} + \sqrt{1-z}$ in a suitable region, and prove that it is analytic. Is my following proof attempt valid? ...
61 views

### $\forall x \in \mathbb{R}$ show that $x=\sum_{n=1}^\infty k_na_n = \prod_{n=1}^{\infty}m_na_n$ …

Yet again, another cool problem from the book "problems in mathematical analysis" by Piotr & Witkowski: Prove that if $a_n \neq 0$, $n=1,2,\cdots$ and $\displaystyle \lim_{n \to \infty} a_n = 0$, ...
31 views

### If $(\sqrt{5}-1)/2 = \sum_{k=1}^{\infty}2^{-n_k}$ where $n_k \in \mathbb{N}$ then $n_k \leq 5\cdot 2^{k-1}-1$

Show that if $(\sqrt{5}-1)/2 = \sum_{k=1}^{\infty}2^{-n_k}$ where $n_k$ are positive integers, then $n_k \leq 5\cdot 2^{k-1}-1$. This is a problem from the book "Problems in mathematicaly analysis" ...
20 views

27 views

### Showing the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$

Problem: If $\sum a_n z^n$ and $\sum b_n z^n$ have radii of convergence $R_1$ and $R_2$, show that the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$. Is the following proof ...
18 views

### Finding the radius of convergence for $\sum n^p z^n$ (Proof Verification)

Goal: Find the radius of convergance for the following complex power series: $$\sum n^p z^n$$ Attempt: We have by Hadamard's formula for the radius of convergence that the complex power series ...
32 views

### Harmonic map into sphere

Let $B$ be the unit ball and $S$ the unit sphere in $\mathbb{R}^3$. Consider the map $u: B\rightarrow S$ defined as: $$u^j(x)=\frac{x_j}{|x|}\quad\forall \ j =1, 2, 3.$$I ...
20 views

### Proof similar to that of Baire's theorem

Sometimes, I read proofs that are very similar to the proof of Baire's theorem, but which I can't directly simplify by using Baire's theorem. I'll give an example: Let $M$ be a complete metric or ...
53 views

### limit proof at a point. Spivak, Chapter 5 problem 36b.

Prove that $\lim_{x\to0^-} f(\frac{1}{x})=L$ iff $\lim_{x\to-\infty} f(x)=L$ Proof $(\leftarrow)$ Suppose that $\lim_{x\to-\infty} f(x)=L$ Let $\epsilon>0$ then there exists a real number ...