Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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About Cantor Set And Measure

Prove that the Cantor ternary set has Jordan content 0. Additionally, prove that the Cantor ternary set has uncountably many points.
user147566
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Is this double integral always positive on nonzero continuous functions?

Is this double integral $$(f,g)=\int_{x=0}^1\int_{y=0}^1\frac{f(x)g(y)}{|y-x|^{\frac14}}dydx$$ an inner product on continuous functions on $[0,1]$? Namely, is $(f,f)$ always positive for all nonzero continuous functions $f$? I don't know if this is…
Stephen
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Solve $\frac{1}{1^z}+\frac{1}{3^z}+\frac{1}{5^z}+\cdots=\frac{1}{2^z}+\frac{1}{4^z}+\frac{1}{6^z}+\cdots$ for $z\in \mathbb C$

My professor gave us this problem. Find all complex numbers $z\in \mathbb C$ such that $$\frac{1}{1^z}+\frac{1}{3^z}+\frac{1}{5^z}+\cdots=\frac{1}{2^z}+\frac{1}{4^z}+\frac{1}{6^z}+\cdots$$ I removed my try because it's wrong.
Superbus
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Connected Subset of Finite Topological Space

What I want to know is this: Am I correct in thinking that if we have a finite topological space (so no reals or anything here), that any connected subset contained in this space has only one element? For example, {x} is connected but {x,y} is…
user138798
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Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$

Prove that for all $x,y>0$ the following inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$ is true. It smells like Jensen inequality, but all I can get is that $\frac{x+y}{2}ln(x) + \frac{x+y}{2} ln(y) \geq xln(\frac{x+y}{2})+yln(\frac{x+y}{2})$
Jules
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Alternatives to Rudin

I'm taking an advanced calculus class this semester and we've been using Rudin's Principles of Mathematical Analysis. I was wondering if anyone could suggest some good analysis textbooks aimed toward undergrad students that I can use as a…
blubberbrot
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Field isomorphism of $\mathbb{C}$ onto itself

I am trying to find a field isomorphism of $\mathbb{C}$ onto itself other than the identity map. The isomorphism preserves the algebraic structures $f(x+y)=f(x)+f(y)$ and $f(xy)=f(x)f(y)$. This means $f(i^2)=f(-1)=f(i)f(i)$. With this in mind, I…
Thiago
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Is $f(x)=\sup_{y\in K}g(x, y)$ a continuous function?

Let $K\subset \mathbb R^n$ be a compact subset and consider a continuous function $g:K\times K\longrightarrow \mathbb R$. Define $f:K\longrightarrow \mathbb R$ by, $$f(x)=\sup_{y\in K}g(x, y).$$ Is $f$ a continuous function?
PtF
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Integration by substitution for line integrals

Let $P, Q: D \rightarrow \mathbb{R}$ be continuous functions on an open set $D \subset \mathbb{R}$, let a curve $\gamma: [a,b] \rightarrow \mathbb{R}^2$ be of class $C^1$ such that $\gamma([a,b]) \subset D$. Let $\Phi=(\phi, \psi): D' \rightarrow D$…
Richard
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When is a function differentiable

Let $f$ be a function $$ f(x,y)=\begin{cases}\quad0&(x,y)=(0,0)\\\dfrac{x^3y^2}{\left(x^2+y^2\right)^2} & (x,y)\neq(0,0). \end{cases}$$ I know what $f$ is continous at the point because the limit of $f$ when $(x,y)\to(0,0)$ exist: did go to the…
Kim
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About real entire function

Assume that $f:(a,b) \rightarrow \mathbf{R}$, $x_0 \in (a,b)$ and $f(x)=\sum_{n=0}^\infty a_n (x-x_0)^n$ for $x \in (a,b)$ and radius of convergence $R$ of this power series is infinite. (Then $f$ is smooth, $a_n=\frac{f^{(n)}(x_0)}{n!}$ for $n=0,…
Richard
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how can I show that the sequence $x_{n}=\frac{n}{n^{2}+1}+\frac{n}{n^2+2}+...+\frac{n}{n^2+n},n=1,2,...$ converges?

How can I check if the sequence $x_{n}=\frac{n}{n^{2}+1}+\frac{n}{n^2+2}+...+\frac{n}{n^2+n},n=1,2,...$ converges?Is there a theorem that I could use?
evinda
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Prove $y = x$ is continuous

For every $\epsilon > 0$ there exists a $\delta > 0$ such that $|x - c| < \delta$ implies $|f(x) - f(c)| < \epsilon$. Start with $|f(x) - f(c)| < \epsilon$ which gives $|x - c| < \epsilon$. We also know $|x - c| < \delta$ but how can we connect…
user2850514
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If I only know $f'(x) = e^{x^2}$, how do I evaluate $\lim_{x \to \infty} \frac{f'(x)}{f(x)}$?

Specifically, I need to show that it equals $\infty$. I remember writing the solution down somewhere, though I can't find it and can't remember it, so I'm mostly looking for an outline of how to solve it. Thanks in advance.
access
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Mathematical indicator for the flatness of a curve

I am currently working on a computer science project where I have to evaluate charts. The charts are simple lines in a $x$ - $y$ coordinate-system, given by CSV files. The flatter the curve, the better for me. Now I am looking for an indicator for…
Thomas Uhrig
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