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).
44,440 questions
3 votes
1 answer
82 views
$f(0)=1$, $f(x) \ge 0 \ge f'(x)$, $f''(x)\le f(x)$ for $x\ge 0$
Problem Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x)$, for all $x\ge 0$. Show ...
- 3,348
-1 votes
2 answers
56 views
Does uniformly continuous functions apply to something like "sandwich theorem"? [closed]
Suppose $f,g$ are two uniformly continuous functions on $\mathbb R$, and $h$ is a continuous function on $\mathbb R$ that satisfies:$$f(x)\le h(x) \le g(x)$$Does that mean $h$ is a uniformly ...
- 9
0 votes
1 answer
106 views
Formal Justification of Alternating Harmonic Series: $1-\frac{1}{2}+\frac{1}{3} - \cdots = \ln(2)$ [duplicate]
It's well known that $\sum_{n=1}^\infty \frac{(-1)^n}{n}=\ln(2)$, which can most easily be seen by the following derivation: $$\frac{1}{1-x} = \sum_{n=0}^\infty x^n \tag{Geometric Series}$$ $$\frac{1}{...
- 5,132
0 votes
0 answers
30 views
Find the best approximant in $L^1$ space to the $0$ function in $[1,-1]$ with weight $w(x) = \frac{1}{\sqrt{1-x^2}}$ [closed]
How can I prove that the best approximant in $L^1$ space to the $0$ function in $[1,-1]$ with weight $w(x) = \frac{1}{\sqrt{1-x^2}}$ is $T_n(x)$, the Chebyshev polynomials of the first kind? I know ...
- 19
1 vote
0 answers
58 views
Representation of this indicator function.
Let $(S,\Sigma)$ be a measurable space, $f_1,\cdots,f_n:S\to\Bbb R$ be measurable functions, and $f:=(f_1,\cdots,f_n)$. For $A\in \mathcal F:=\{f^{-1}(B)\mid B\in\mathcal B(\Bbb R^n)\}$, I want to ...
- 340
3 votes
2 answers
404 views
The intuition behind defining trigonometric functions in the complex plane as special combinations of exponential functions [closed]
I was studying Complex Analysis from "A First Course of Complex Analysis" and the authors stated directly that sine and cosine are defined as follows (without any intuition): $$ \sin\left(z\...
- 31
0 votes
0 answers
53 views
On computing the infimum of a functional [closed]
Let $P, Q \in \mathbb{R} \to \mathbb{R}$ be monic polynomials with real coefficients such that $\deg(Q) > \deg(P)$, $\min P > 0$, $\min Q > 0$. For $a<b$ and $A,B\in\mathbb{R}$, define $$ ...
- 83
1 vote
1 answer
62 views
upper bound of the modulus of the complex-valued function $F(z)=\frac{1+i\theta(z)}{1-i\theta(z)}$
This little problem arises in a proof of a generalization of the Hermite-Biehler theorem. Let $$ F(z):=\frac{1+i\theta(z)}{1-i\theta(z)}, $$ where $\theta(z)$ maps the upper half-plane onto the upper ...
- 195