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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Prove that a function is differentiable on $\mathbb R$

Problem: Let $f: \mathbb R -> \mathbb R$ be given by $f(x):= x.|x|$. Show that $f$ is continuous and differentiable on $\mathbb R$. My solution: By using the Differentialquotient we prove that $f$ is differentiable $ lim_{x\to x_0} = \left(\frac{…
Kai
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If there exists a positive $K$ such that $|f(x)| \leq K \int_a^x |f(t)|dt$ then $f(x) = 0$

Let $f$ is continuous on $[a,b]$. There exists positive K such that $|f(x)| \leq K \int_a^x |f(t)|dt$ then $f(x) = 0$. I was trying to prove the statement above, by trying the smallest number c such that $f(x) = 0$ for any $x
LemonTea
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arc length of helix $(2\cos(t), 2\sin(t), 3t)$

Hey i have to calculate the arc length of a helix $f:[0,2πn]\to R^3: t\mapsto(2\cos(t), 2\sin(t), 3t)$ I think I have to calculate the arc length as an integral over norm of speed vector. So I started like this: $$f'(t)=(-2\sin(t), 2\cos(t),…
user786835
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how to find the sum for $\sum_{k=2}^{\infty} k(k-1)x^{k-2}$?

I know that the convergence radius is $R = (-1,1)$, but I don't know how to sum this series, please help.
Indr
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Does $f_{n}(x)=\frac{x^{2n}}{1+x^{2n}}$ converge pointwise / uniformly?

Since our lectures were cancelled because of the ongoing situation, I have to essentially self-study for my analysis exam in two months. Understandably this comes with a great deal of trouble, so I would like someone to help me with the following…
average math enjoyer
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Differentiable implies continuity

I am slightly confused about the epsilon-delta proof of this theorem. I fully understand up to the point that: $$|f(x)-f(a)| < (|′()|+)⋅|−|$$ We then pick delta to be the $\hat{} =min(,̂/ (|′()|+))$ Thus, $|x-a|< \hat{}$. So if $$|f(x)-f(a)| <…
jeff123
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$\int^x_2 (\log u)^{-2} du\ll x(\log x)^{-2}$

I came across this estimation in a book: $$ \int^x_2 (\log u)^{-2} du \ll x(\log x)^{-2}. $$ I tried to prove it by first integrating by parts: $$ \int^x_2 (\log u)^{-2} du = x(\log x)^{-2}-2(\log 2)^{-2} + 2\int^x_2 (\log u)^{-3} du, $$ but was…
Tapioka
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Integral of $\arccos$ within $\sin$

How to calculate the following integral: $\int^{R}_{0}[2 \cos^{-1}(\frac{r}{2R}) -\sin(2 \cos^{-1}(\frac{r}{2R}) ) ] dr$. This is a part of a complex formula.
Noureddine
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Zeros of real exponential polynominals

How can I solve an equation like $ x^a + bx + c = 0 $ ? I figured that I can write this equation in a polynominal form as $ e^{wz} + b e^{z} + c $ or more generic as $ \sum{v_i e^{w_i z}} $ and that these are called exponential polynominals. There…
Phaiax
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Sum of rational functions with exponent

How to compute $$\sum_{j=1}^{K}\frac{P(j)}{Q(j)}\exp(2\pi ija) $$ where $\left|a\right|<1,\ K\in \mathbb{Z}$, $\frac{P(j)}{Q(j)}$ is a rational function and the roots $Q(j)$ are known, complex.$$$$ In my case…
Katja
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Determine the limit of the expression $\frac {\sum_{k=1}^n \frac{1}{k} - (1-\frac{2}{n})\ln n}{\ln n}$ as $n \to \infty$

I would like to analyze the convergence of the expression $$a_n =\frac {\sum_{k=1}^n \frac{1}{k} - (1-\frac{2}{n})\ln n}{\ln n}$$ as $n \to \infty$. In particular, I'd like to know that, if the limit exists and is finite, whether it is $0$. The…
JZS
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$f$ is a bijective function with differentiable inverse at a single point

Let $\Omega \subseteq \mathbb{R}^n$ and $p \in \Omega$. Let $f:U \to V$ be a bijection of open sets $p \in U \subseteq \Omega$ amd $f(p) \in V \subseteq \mathbb{R}^n$. If $f^{-1}: V \to U$ is differentiable at $p$, then $df_p: \mathbb{R}^n \to…
emka
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Non-negative orthant: smoothing

Consider the non-negative orthant $$ R\equiv \{(x_1,...,x_n): x_i\geq 0 \text{ }\forall i\} $$ The boundary of $R$ is not smooth. I'm looking for a function which can smooth the boundary of $R$ and which depends on only one smoothing parameter.…
Star
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Let $(E, A, \mu)$ be a measure space, $g: E \to [0, +\infty)$ be a positive measurable function.

$v: \mathcal{A} \to [0, +\infty]$ by $v: A \mapsto \int_A g d\mu$ (a) Let $f: E \to [0, +\infty)$ be positive and measurable. Show that (*)$$\int_E f dv = \int_E g d\mu$$ (b) Show that f is $v$-integrable if and only if $fg$ is $\mu$-integrable,…
Serena sss
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What’s wrong with this proof that the set of natural numbers is uncountable?

Suppose the set is countable, and can be arranged in a sequence. Then we construct a number whose nth digit is different than the nth digit of the nth number in the sequence, which means that it is not in the sequence.
John
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