Intereting Posts

$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$
Is it possible to make integers a field?
On the number of caterpillars
integration as limit of a sum
Some Kind of Generalized Brownian Bridge
Value of $\sum_{i=1}^{p} i^k \pmod{p}$
Show that $\lim\limits_{n \to \infty} \sup\limits_{k \geq n} \left(\frac{1+a_{k+1}}{a_k}\right)^k \ge e$ for any positive sequence $\{a_n\}$
A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$
Roots of a polynomial whose coefficients are ratios of binomial coefficients
What is the kernel of $K \to K$, defined by $T \mapsto x$?
If $p\in R$ is irreducible, is it still irreducible in $R$?
“This statement is false” – Propositional Logic
Bhattacharya Distance (or A Measure of Similarity) — On Matrices with Different Dimensions
Comparing the sizes of countable infinite sets
Is there a slowest divergent function?

i dunno if this is asked before, and i am not sure where to find this on the web or in textbooks.

we are given a function (that is too hard to invert by solving for $x$):

$$ y = f(x) $$

- $\int_0^{\pi}{x \over{a^2\cos^2 x+b^2\sin^2 x}}dx$
- Closed form: $\int_0^\pi \left( \frac{2 + 2\cos(x) -\cos((k-\frac{1}{2})x) -2\cos((k+\frac{1}{2})x) - cos((k+\frac{3}{2})x)}{1-\cos(2x)}\right)dx $
- What is the integral of 0?
- Evaluate $\int_{0}^{+\infty }{\left( \frac{x}{{{\text{e}}^{x}}-{{\text{e}}^{-x}}}-\frac{1}{2} \right)\frac{1}{{{x}^{2}}}\text{d}x}$
- Find a continuous function $f:[1,\infty)\to\Bbb R $ such that $f(x) >0 $, $\int_1^\infty f(x)\,dx $ converges and $\int_1^\infty f(x)^2\,dx$ diverges
- Equivalent Cauchy sequences.

which has an unknown inverse:

$$ x = g(y) $$

so $ y = f(g(y)) $ and $ x = g(f(x)) $ and let’s say that $f(\cdot)$ is an odd-symmetry function

$$ f(-x) = -f(x) \quad \quad \forall x \in \mathbb{R} $$

i think that means that $g(\cdot)$ must also be an odd-symmetry function.

since we know $f(x)$, we can compute derivatives of it around $x=0$. we can represent $f(x)$ as a Maclaurin series. and we also know that all of the even-power terms of the series are zero.

$$ y = f(x) = x \cdot \sum\limits_{n=0}^{\infty} a_n \ x^{2n} $$

the same can be said about $g(y)$

$$ x = g(y) = y \cdot \sum\limits_{n=0}^{\infty} b_n \ y^{2n} $$

i know coefficients $a_n$ because i know $f(x)$ and all of its derivatives. now by slugging through this manually, i can compute coefficients $b_n$ for the Maclaurin expansion of $g(y)$. the first two are

$$ b_0 = \frac{1}{a_0} $$

$$ b_1 = \frac{-a_1}{a_0^4} $$

now, is there a more general method of computing these inverse coefficients? is there a paper or online reference that deals with this, ostensibly quite practical, calculus problem? or am i sentenced to just slog through this manually and stop at the $y^5$ or $y^7$ term?

- Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$
- Derivative of a product and derivative of quotient of functions theorem: I don't understand its proof
- $\int_{-\infty}^{\infty}{e^x+1\over (e^x-x+1)^2+\pi^2}\mathrm dx=\int_{-\infty}^{\infty}{e^x+1\over (e^x+x+1)^2+\pi^2}\mathrm dx=1$
- Prove that if $\alpha, \beta, \gamma$ are angles in triangle, then $(tan(\frac{\alpha}{2}))^2+(tan(\frac{\beta}{2}))^2+(tan(\frac{\gamma}{2}))^2\geq1$
- Limit of a continuous function
- $\frac{1}{\infty}$ - is this equal $0$?
- Dedekind's cut and axioms
- Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$
- Are the any non-trivial functions where $f(x)=f'(x)$ not of the form $Ae^x$
- Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

- Good book for convergence of series
- On the solution of constant coefficients PDEs (exponential method)
- Polynomial Question
- Do “other” trigonometric functions than Tan Sin Cos and their derivatives exist?
- Minimal polynomial for an invertible matrix and its determinant
- An integral to prove that $\log(2n+1) \ge H_n$
- How to deduce the area of sphere in polar coordinates?
- Prove by elementary methods: the plane cannot be covered by countably many copies of the letter “Y”
- Is there any diffeomorphism from A to B that $f(A)=B$?
- Improper Riemann integral of bounded function is proper integral
- Example of non-homeomorphic spaces $X$ and $Y$ such that $X^2$ and $Y^2$ are homeomorphic
- a Fourier transform (sinc)
- Fastest way to check if $x^y > y^x$?
- What is the value of the integral$\int_{0}^{+\infty} \frac{1-\cos t}{t} \, e^{-t} \, \mathrm{d}t$?
- Entire function. Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$