Questions about studying mathematics without formal instruction.

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2
votes
1answer
62 views

Derive a perturbation of period $2\pi$, to order $\epsilon$

I have the following problem: In the equation $\ddot{x}+\Omega^2x+\epsilon f(x) = \Gamma \cos t$, $\Omega$ is not close to an odd integer, and $f(x)$ is an odd function of x, with expansion, $$f(a\cos ...
2
votes
3answers
98 views

Gamma function can be extended as a meromorphic function

Let $$\Gamma (z)= \int_{0}^{\infty} e^{-t}t^{z-1}dz$$ for $\Re z\gt 0$ Can be extended as a meromorphic function to the entire complex plane with simple poles at non positive integers. How can I ...
0
votes
1answer
26 views

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$ I know the result. But I dont know how to show this step by step.
-1
votes
1answer
45 views

Learning way and Resource for Complete math Subject. [closed]

I want to learn [self learning] Mathematics from basic.What is the order [like 1) arithmetic,2) Geometry,Etc..] to learn the maths? and what is the best resource to that particular subject?
2
votes
4answers
63 views

What does countable union mean?

The book I am reading contains the following two definitions: Two sets $A$ and $B$ have the same cardinality if there exists $f: A \rightarrow B$ that is one to one and onto. In this case, we ...
6
votes
1answer
463 views

What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
1
vote
1answer
144 views

How to linearize a nonlinear ODE around its equilibrium?

I am studying for a comprehensive exam in non-linear ODE's and I have this in my book: $$\ddot{\xi}+c\bigg[x_1+\xi-\dfrac{\lambda}{a-x_1-\xi}\bigg] = 0$$ then it goes straight to ...
7
votes
1answer
94 views

Find $\sum_{k=1}^{\infty}\frac{1}{z_k^2}$

Let $z_1, z_2,\dots, z_k,\dots$ be all the roots of $e^z=z$. Let $C_N$ be the square in the plane centered at the origin with siden parallel to the axis and each of length $2\pi N$. Assume that ...
0
votes
1answer
63 views

Why can't the interval construction argument used to show $\mathbb{R}$ is uncountable be used for other infinite sets?

I read the following proof as to why the set of real numbers is uncountable. Assume that $\mathbb{R}$ is countable. Then we can enumerate $\mathbb{R} = \{x_1, x_2, x_3, \cdots\}$ and be sure that ...
1
vote
1answer
52 views

Zeta function in complex analysis.

Show that $$\frac{\zeta'(z)}{\zeta(z)}=-\sum_{n=2}^{\infty}\frac{f(z)}{n^z}$$ for $\Re z\gt 1$ Where $f(z)= \ln p$ if $n=p^m$ for some prime $p$ and some $m\in \Bbb N^+$ Or $f(z)=0$ otherwise. ...
4
votes
2answers
205 views

Exercise Real Analysis

I'm having trouble to understand the following exercise I would appreciate any help? Let $A,B,C$ be sets such that $A\subseteq B\subseteq C$ and let $f:C\rightarrow A$ be an injective map. Define ...
2
votes
1answer
56 views

Will a bounded sequence in $\mathbb{R}$ necessarily induce a compact subset of $\mathbb{R}$?

Question: We know that a bounded sequence in $\mathbb{R}$ must have a convergent sub-sequence from Bolzano-Weierstrass theorem. However, will a bounded sequence $(x_n)$ in $\mathbb{R}$ necessarily ...
7
votes
3answers
2k views

Functional analysis textbook (or course) with complete solutions to exercises

I am a Ph.D. student in economics and I plan to study functional analysis by myself either this winter or the next summer. I am currently looking for a textbook, and since I am studying it by myself, ...
2
votes
2answers
185 views

How to handle dependence in a probabilistic ball problem?

I am learning about how to handle dependence when doing probabilistic calculations and I came across this problem I don't understand how to solve. I am given $10$ colored balls painted either red or ...
1
vote
1answer
44 views

Boundary Value Problem (Separation of Variables)

Solve the boundary value problem \begin{cases} u_{t}-2u_{xx}=0 \\ u_{x}(0,t)=u_{x}(\pi,t)=0, \quad x\in[0,\pi], t\geq0 \\ u(x,0)=\cos^{2}(x) \end{cases} My Attempt Let $u(x,t)=X(x)T(t)$. Then ...
1
vote
1answer
64 views

Why does $\operatorname{rank}(\mathbf{X})$ equal $\operatorname{rank}(\mathbf{X^TX})$? What is $\dim(\mathbf{X^TX})$?

Why does $\operatorname{rank}(\mathbf{X})$ equal $\operatorname{rank}(\mathbf{X^TX})$? Is this true in general, please? And what is $\dim(\mathbf{X^TX})$, please? Does it equal to $\dim(\mathbf{X})$ ...
2
votes
3answers
100 views

Derivative of a determinant whose entries are functions

Happy New Year, everyone! I do not understand a remark in Adams' Calculus (page 628 $7^{th}$ edition). This remark is about the derivative of a determinant whose entries are functions as quoted below. ...
211
votes
22answers
12k views

On “familiarity” (or How to avoid “going down the Math Rabbit Hole”?)

Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole". For example, suppose you come across the novel term vector space, and want to learn more ...
10
votes
1answer
300 views

Preparing for “differential forms in algebraic topology”?

I'd very much like to read "differential forms in algebraic topology". Apart from background in calculus and linear algbra I've thoroughly went through the first 5 chapters of Munkres. I'm thinking ...
3
votes
1answer
85 views

Branch points of global analytic functions (Ahlfors)

In Ahlfors' complex analysis text, page 298 he discusses the case of a global analytic function $\mathbf{f}$ which can be continued analytically along all arcs in some punctured disk $\{0<|z|< ...
39
votes
8answers
5k views

When to learn category theory?

I'm a undergraduate who wishes to learn category theory but I only have basic knowledge of linear algebra and set theory, I've also had a short course on number theory which used some basic concepts ...
1
vote
1answer
65 views

Rearrangement Thm

Hi everyone: In the book what I've read of analysis in the proof of the Riemann rearrangement thm there is gaps that I need to fill. There is no real problem in almost everything but for the last one ...
3
votes
1answer
142 views

What are the benefits or losses of learning real analysis through a constructivist approach instead of a standard apporach?

Recently I've found some courses on real analysis that use the constructivist approach and I got curious on some aspects: What are the benefits of learning through this approach? Is it ok to learn ...
2
votes
1answer
130 views

Naturality conditions.

I'm following Turi's Category Theory Lecture Notes, which can be found online. I'll try to make downloading a copy unnecessary. Here's my problem. Definition: An adjunction $\langle F\dashv U, ...
1
vote
2answers
84 views

Infinite product convergence

Prove that $$\prod_n\left(1+\frac{i}{n}\right)$$ diverges. But $$\prod_n\left\vert 1+\frac{i}{n}\right\vert$$ converges. I know the theorem $\prod (1+z_k) $ converges $\iff$ $\sum\log (1+z_k)$ ...
1
vote
1answer
66 views

Where am I wrong applying chain rule here?

Why is $$\dfrac{\partial}{\partial x_i}f(tx_1, \dots, tx_n)=\dfrac{\partial f}{\partial(tx_i)}(tx_1,\dots,tx_n)\cdot\dfrac{\partial(tx_i)}{\partial x_i}=\dfrac{\partial f}{\partial ...
1
vote
1answer
42 views

Basic concerns about dependent function types

I am working my way through the introductory material of Homotopy Type Theory and by the end of section $1.5$ it is clear that I did not have the earlier material down as clearly as I had thought. I ...
0
votes
2answers
85 views

Difference of the Cauchy Sequences is Cauchy

Assume $\{a_n\}\;$ and $\;\{b_n\}\;$ are Cauchy sequences. Use a triangle Inequality Argument to prove $\{c_n=|a_n-b_n|\}\;$ is Cauchy. So in the answer key, the author proved it by taking ...
2
votes
3answers
149 views

Learning math for physics

I am very interested in physics and am planning to self studying it. But for this I need to be mature in various areas of math. So I want to know what is the order in which I need to learn the math ...
2
votes
1answer
98 views

Prove that $\Gamma'(1)=-\gamma$

Use the product formula for $1/\Gamma(z)$ to prove that $$\Gamma'(1)=-\gamma$$ I know that for Euler constant $\gamma$, $$\frac{1}{\Gamma(z)} =ze^{\gamma z}\prod _{k=1}^{\infty} ...
1
vote
2answers
144 views

Gamma function in complex analysis.

Prove that $$ \Gamma\left(z\right) = \lim_{n\to \infty}\int_{0}^{n}t^{z - 1}\left(1 - {t \over n}\right)^{n}\,{\rm d}t \quad\mbox{for}\quad \Re z \gt 0 $$ I know that $$ {\rm e}^{-t/n} = 1 - {t ...
1
vote
4answers
163 views

The complex gamma function

Show that $$\Gamma (z+1)=z\Gamma (z)$$ $\forall z\in \Bbb C$ except for $z=-n$ where $n\in \Bbb N$. I know that the gamma function is defined as $\Gamma (z)=\int_{0}^{\infty}e^{-t}t^{z-1}dt$ And ...
2
votes
0answers
53 views

Counter-Example for $f$ unbounded, $f$ continuous at $s$, and $\alpha(x)=I(x-s)$ such that $\int_a^b f \,d\alpha \neq f(s)$ (Rudin Thm 6.15)

I'm reading Theorem 6.15 in Baby Rudin, which states that: Theorem 6.15: If $a<s<b$, $f$ is bounded on $[a,b]$, $f$ is continuous at $s$, and $\alpha(x)=I(x-s)$, then $$\int_a^b f \,d\alpha = ...
11
votes
8answers
3k views

A Book for abstract Algebra

I am self learning abstract algebra. I am using the book Algebra by Serge Lang. The book has different definitions for some algebraic structures. (For example, according to that book rings are defined ...
-1
votes
2answers
74 views

What textbook and chapters should I go through to study for basic math qualification exam?

In 7 months I will have to take a "Basic Math" qualification exam similar to the ones shown here: Fall 2013 Spring 2013 Fall 2012 Spring 2012 Fall 2011 Could somebody please recommend the best ...
0
votes
2answers
81 views

What textbook and chapters should I go through to study for a probability qualification exam?

In 7 months I will have to take a "Probability" qualification exam similar to the ones shown here: Fall 2013 Spring 2013 Fall 2012 Spring 2012 Fall 2011 Could somebody please recommend the best ...
7
votes
4answers
272 views

Advanced Mathematics

I am a high school student and would like to pursue a career in mathematics and I am hoping to find a serious explanatory book on math (geometry, algebra, calculus, functions and trigonometry) for ...
2
votes
2answers
179 views

What's the right moment to learn Set Theory?

I've seen a question in which the OP asked when is the right moment to learn Category Theory, it seems this moment comes a little after a course of algebra, and indeed some books on abstract algebra ...
6
votes
1answer
133 views

Soft Question: Suggestions on mathematics resources for problem solving.

I'm doing my final year of under graduation through distance education and would be appearing for entrance tests for various graduate schools in a few weeks. I am looking for a database of ...
3
votes
1answer
160 views

Finite order function in the complex analysis.

Assume that an entire function $f$ be finite order with finitely many zeros. Please show that either $f(z)$ is a polynomial or $f(z) + z$ has infinitely many zeros. Thank you. And I know the ...
2
votes
0answers
99 views

Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
11
votes
2answers
133 views

Is the maximal path through a math book necessarily linear?

I'm studying with two main math books (Munkres and D&F) these couple of months. My method so far is just going through the book page by page constructing everything in it (independently if I can) ...
5
votes
1answer
127 views

Prerequisites for bredon's “Topology and Geometry”?

My background in topology is the first 6 chapters of Munkres's "topology" and in algebra Herstein's "topics in algebra". Both of them I self studied. A look at the table of contents of bredon's ...
3
votes
0answers
89 views

Learn enumerative combinatorics? [closed]

I am interested in becoming proficient in enumerative combinatorics relatively quickly. I want to be able to look at a problem briefly and think of multiple different useful approaches to it. Any ...
0
votes
2answers
749 views

How to learn calculus for beginners? [duplicate]

As a precalculus student interested in teaching myself calculus, where should I start and how should I go about learning? This question is different than past questions as I am not solely interested ...
0
votes
1answer
56 views

Probability of getting same number of black and white pebbles

I am trying to learn some probability and the following problem summarises my current confusion quite well. Any help is gratefully received. I have $n$ pebbles, some of which are black, some white ...
2
votes
0answers
170 views

Fubini's Theorem for Infinite series

In the book what I've read, there is one point where the author suggest to begin the proof of the Fubini's Theorem for infinite sum in the case when is non-negative after this try to generalize. But ...
13
votes
2answers
296 views

Being mathematically critical: how should a student approach statements that appear to be obvious?

Very occasionally, I will read or hear a theorem, and think: isn't that obvious? Not in a contemptuous "I can immediately see how to prove this" way, but rather in a "I would have thought this was ...
1
vote
1answer
140 views

$f(z)$ has infinitely many zeros and that each zero is simple.

Let $f(z)=e^z-z$ I want to check $f(z)$ is finite order. And how to show that $f(z)$ has infinitely many zeros and that each zero is simple. Dfn: an entire function f is finite order if ...