# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

2,544 questions

**0**

votes

**0**answers

41 views

### A hard to answer Digamma question [on hold]

I just learning online obout Polygamma function, and I want to know what x equal to (and how to get it) when ψ(x)=1 and x>1.

**-2**

votes

**0**answers

40 views

### What is the relationship (mapping) from a reciprocal function 1/r to a exponential function exp(-r)? [on hold]

The mathematical problem:
Consider the mapping from $r$ to $u$.
for large $r$ the theory suggests a formulation like $u=a_1 e^{-a_2 r}$, which means that the function decay exponentially.
for ...

**1**

vote

**0**answers

53 views

### How to see the divergence of a series is not faster than some order? [on hold]

$$
\sum_{m=1}^{n} m^{-1+1/p} \leq Cn^{1/p}
$$
For $1<p<2$, I know the LHS is divergent but I can't see its speed of divergence is not faster than $n^{1/p}$.

**5**

votes

**0**answers

70 views

### Square of Jacobian

Given a smooth vector-valued function $u:B_1(0)\subset \mathbb{R}^n\to \mathbb{R}^m$, one can look at the matrix
\begin{equation}
Q_{ij}= \sum_{k=1}^m \nabla_i u^k \nabla_j u^k
\end{equation}
This is ...

**-3**

votes

**0**answers

34 views

### How to find the volume using triple integral spherical coordinates? [on hold]

A hemispherical bowl of radius $5 cm$ is filled with water to within $3cm$ of the top. Find the volume of water in the bowl?
How can I find the volume using spherical coordinates?
writing it like ...

**-1**

votes

**0**answers

70 views

### Counter-example of Subsequence Criterion? [migrated]

The last argument shows that if $X_n\to X_\infty$ a.s. and $N(n)\to\infty$ a.s., then $X_{N(n)}\to X_\infty$.
We have written this out with care because the analogous result for convergence in ...

**-2**

votes

**0**answers

40 views

### Closed union of all connected subsets that contain x [on hold]

If Cx(S) is the union of all connected subsets of S which contain x, it is connected. I understand that, but what I don’t understand is that if S is closed, then Cx(S) is closed. Isn’t that like ...

**2**

votes

**2**answers

99 views

### Existence of solution to linear fractional equation

We consider the equation
$$?\sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...

**4**

votes

**0**answers

93 views

### Approximation of a compactly supported function by Gaussians

Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...

**10**

votes

**1**answer

315 views

+50

### A question concerning Lusin’s Theorem

We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$.
Let $f$ be measurable. For every $e$ in $...

**4**

votes

**0**answers

166 views

### A simple proof of Jordan curve theorem [closed]

I need a short proff of the Jordan curve theorem please.
The one I have is 16 pages long and is for a little expo, so I need one a little shorter.
Thanks

**3**

votes

**1**answer

77 views

### Equivalent notion of approximate differentiability

Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one?
$$\lim_{r \to 0} \rlap{-}\!\!\int_{...

**-1**

votes

**0**answers

37 views

### Baseline measurements [closed]

I need to compare 2 different results to determine best performance (best defined as lowest).
Baseline measure is set for each level for expected performance.
Baseline measure for each level ...

**1**

vote

**0**answers

33 views

### Formal justification of the Chaos game in the Sierpinsky triangle

I want to justify why the Chaos game works to produce Sierpinsky triangle. I use a theorem taken from Massopoust Interpolation and Approximation with Splines and Fractals.
Suppose that $(X,d)$ is a ...

**2**

votes

**0**answers

41 views

### Generalized definition of integrable condition on rough complex subbundle

Assume object are smooth at first. If we consider real subbundle, we can define integrability in terms of parameterization or coordinate.
A rank $r$ real subbundle $\mathcal V\le TM$ is called ...