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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The convergence of $\sum_1^{+\infty}b_n$ follows from the convergence of $\sum_1^{+\infty}a_n$ given that $\frac{a_n}{b_n}\to1$.

It is true that if $$ \sum_1^{+\infty}a_n\qquad\text{and}\sum_1^{+\infty}b_n $$ satisfies $$ \lim_{n\to+\infty}\frac{a_n}{b_n}=1, $$ then the convergence of $\sum_1^{+\infty}b_n$ follows from the convergence of…
Knt
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How much pure math should a physics/microelectronics person know

I do condensed matter physics modeling in my phd and I was struck up learning quite an amount of physics. But while having done lot of physics courses, I see that if I learn pure math I would understand lot of things better and also probably my…
pencil
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Is $f$ constant if every point is local maximum or local minimum of $f$?

Suppose $f: I \to \mathbb{R}$ where $I = (a,b)$ or $I = \mathbb{R}$, etc. Suppose that every $x$ is either a local maximum of $f$ or a local minimum of $f$. Does it follow that $f$ is a constant function? I think it's probably easy if you know $f$…
Silas Bunion
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What are some examples of non smooth continuous functions without "kinks"?

In a recent post I asked what are the ways a continuous function cannot be smooth. Apparently my question was not well posed, but since I am still thinking about these types of functions I will try one more time to ask my question in a proper…
Matt Calhoun
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I have to think of a metric that makes (0,1) an unbounded interval

I don't even know how to begin to answer this question?
user 85795
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Spherical mean property

Let $ u(x)$ a continuos function over a domain $\Omega$. Let $N\omega_n r^{N-1}$ the area of sphere in $R^N$. I don't understand the reason of this limit: $$ \dfrac{1}{N \omega_n \epsilon^{N-1}} \int_{\partial B_{\epsilon}(y)}u(x) d\sigma …
Giovanni Febbraro
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Proof that a sequence is Cauchy.

Show that $\left( x_{n}\right) $ is a Cauchy sequence, where $$ x_{n}=\frac{\sin1}{2}+\frac{\sin2}{2^{2}}+\ldots+\frac{\sin n}{2^{n}}. $$ We try to evaluate $\left\vert x_{n+p}-x_{n}\right\vert $ and to show that this one is arbitrary small.…
stefano
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Proof that Laplacian is surjective $\mathcal{P}^n\to\mathcal{P}^{n-2}$

Let $\mathcal{P}^n$ denote the vector space of homogeneous polynomials on $\mathbb{R}^3$ of degree $n$. I need to prove that $\Delta|_{\mathcal{P}^n}:\mathcal{P}_n\to\mathcal{P}_{n-2}$, for $n\geq2$ is surjective, where $\Delta$ is the Laplace…
sonjcy
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Finding zeroes of given function

If $f(x)$ is a twice differentiable function, continuous in it's domain such that $f(a)=0$, $f(b)=2$, $f(c)=-1$, $f(d)=2$ and $f(e)=0$ where $a
darkv0id
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Exercise on derivatives from Rudin's Principles of Mathematical Analysis

Exercise 5.26 --Rudin [Principle of Mathematical Analysis] Ex.5.26 If $f'$ exists on $[a,b]$, $f(a)=0$ and $\exists A\in\mathbb{R}\;(|f'|\le A|f|\,\text{on }[a,b])$, then $f = 0$ on $[a,b]$. Hint: Fix $x_0\in [a,b]$, let $M_0 = \sup |f|([a,x_0])$,…
docholliday
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Prove that $1/f$ is Riemann integrable on $[a,b]$

Prove that if $f$ is Riemann integrable on $[a,b]$ and $0<\rho \le f(x)$ for all $x \in [a,b]$ then $1/f$ is Riemann integrable on $[a,b]$.
landolf
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Bound of a function $f_n=\frac{x^2}{x^2+(1-nx)^2}$

Let $\displaystyle f_n=\frac{x^2}{x^2+(1-nx)^2}$ where ($0\le x\le 1,\; n=1,2,3,...$) Then $|f_n(x)| \le M$. Find this $M$. The answer is $1$. Without any restriction of n, how can we find that bound?
neoderm
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Diffeomorphism on the Sphere

Suppose $f:S^2\rightarrow S^2$ is a diffeomorphism. Is true that $f$ must have a fixed point or a point $x\in S^2$ such that $f(f(x))=x$?
Tomás
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Find a bijection from $[0,1]$ to $[0,1]$ that is not strictly monotone. is this possible?

I'm not convinced this is possible, as soon as you have $2$ distinct elements mapping to the same number, the function is no longer $1$-$1$ and therefore not a bijection.
user480172
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Approximation of exponential function by power series

Let $x \in (-\frac{1}{2},\frac{1}{2}), n \in \mathbb{N}$ How can I choose a $n$ that the the inequality is valid? $$\left|e^x-\sum_{k=0}^n \frac{x^k}{k!}\right| \leq \frac{|e^x|}{10^{16}}$$ My ideas: Try some values for $n$ and verify the inequality…
Marc
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