Intereting Posts

Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
The preorder of countable order types
Are there fewer positive integers than all integers?
What is the general equation equation for rotated ellipsoid?
Connected But Not Path-Connected?
Show that $\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx =-\frac{\pi^2 \sqrt{2}}{16}$
Any N dimensional manifold as a boundary of some N+1 dimensional manifold?
Integrating $\frac{1}{1+z^3}$ over a wedge to compute $\int_0^\infty \frac{dx}{1+x^3}$.
How to solve this Complex inequality system
What's the difference between isomorphism and homeomorphism?
Smallest graph with automorphism group the quaternion $8$-group, $Q_8$
Evaluation of $\int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$
How to prove this inequalities involving measures
Orthonormal Matrices-Intuition
For any sets $A$ and $B$, show that $(B\smallsetminus A)\cup A=B \iff A\subseteq B$.

I’m not really sure how to go about this.. any help is appreciated.

- How is the Taylor expansion for $f(x + h)$ derived?
- Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?
- What is the 90th derivative of $\cos(x^5)$ where x = 0?
- How to calculate square root or cube root?
- Help with this limit?
- Method of Steepest Descent and Lagrange
- Is the following scheme for generating $p_n=(1/3)^n$ stable or not. $p_n=(5/6)p_{n-1}-(1/6)p_{n-2}$.
- What are the properties of the roots of the incomplete/finite exponential series?
- Generalized binomial theorem
- Finding all roots of polynomial system (numerically)

Apply the extended mean value formula several times to

$$

\frac{-3f(x)+4f(x+h)-f(x+2h)-2hf'(x)}{h^3}

$$

$$

=\frac{4f'(x+h_1)-2f'(x+2h_1)-2f'(x)}{3h_1^2}\\

=\frac{4f”(x+h_2)-4f”(x+2h_2)}{6h_2}\\

=\frac{4f”'(x+h_2+h_3)}{6}\\

$$

where $0<h_3<h_2<h_1<h$.

Of course one can also do this as Taylor series. Consider

$$g(h)=-3f(x)+4f(x+h)-f(x+2h).$$

Then $g(0)=0$, $g'(0)=2f'(x)$, $g”(0)=0$, $g”'(0)=-4f”'(x)$, thus

$$g(h)=2h·f'(x)-\frac23h^3·(2f”'(x+2h_1)-f”'(x+h_1)).$$

For fixed $h$, define the finite difference operator $\Delta$ as:

$$\Delta f(x) = f(x+h) – f(x)$$

We can then write:

$$f(x+h) = \left(1+\Delta\right)f(x)$$

The Taylor series also gives us an expression for $f(x+h)$ in terms of $f(x)$, which can be written in the compact form:

$$f(x+h)= \exp\left(hD\right)f(x)$$

where $D$ is the differential operator. Comparing the expressions for $f(x+h)$ in terms of the finite difference operator and the differential operator yields identity:

$$D = \frac{1}{h}\log\left(1+\Delta\right)$$

Expanding in powers of $\Delta$ till order $\Delta^2$ yields:

$$D \approx \frac{1}{h}\left(\Delta – \frac{\Delta^2}{2}\right)$$

To calculate the action of powers of $\Delta$ on $f(x)$, we write:

$$\Delta = E-1$$

where the operator $E$ is defined as:

$$E f(x) = f(x+h)$$

Therefore:

$$\Delta^n = \left(1+E\right)^n = \sum_{k=0}^n\binom{n}{k}(-1)^{n-k}E^k$$

So, we have:

$$\begin{split}

\Delta f(x) &= f(x+h) – f(x)\\

\Delta^2 f(x) &= f(x+2h) – 2 f(x+h) + f(x)

\end{split}

$$

This then yields the approximation:

$$D f(x)\approx \frac{1}{h}\left( – \frac{3}{2}f(x) + 2 f(x+h)-\frac{1}{2} f(x+2h)\right)$$

- Is the set $P^{-1}(\{0\})$ a set of measure zero for any multivariate polynomial?
- ${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$
- Ellipsoid but not quite
- Prove $GL_2(\mathbb{Z}/2\mathbb{Z})$ is isomorphic to $S_3$
- Codimension 1 homology represented by Embedded Submanifold
- Pointwise convergence of a sequence of functions
- If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?
- The set of all infinite binary sequences
- Induction proof. Explain in detail why it’s incorrect
- Associativity of Cartesian Product
- Uniformly distributed rationals
- Integration of forms and integration on a measure space
- Is the group of units of a finite ring cyclic?
- Why is the number of possible subsequences $2^n$?
- Compact metrizable space has a countable basis (Munkres Topology)