Intereting Posts

Mathematical properties of two dimensional projection of three dimensional rotated object
How to test if a graph is fully connected and finding isolated graphs from an adjacency matrix
Centralizer/Normalizer of abelian subgroup of a finite simple group
Stochastic Integrals are confusing me; Please explain how to compute $\int W_sdW_s$ for example
Exercise books in linear algebra and geometry
Chebyshev Inequality for Martingales
Conditional convergence of Riemann's $\zeta$'s series
Artinian affine $K$-algebra
Hopkins-Levitzki: an uncanny asymmetry?
Why does Wolfram Alpha say the roots of a cubic involve square roots of negative numbers, when all three roots are real?
A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x^{(n)} dx$
Prove that the centralizer is a normal subgroup.
Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space
Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
How to determine the series for $ f(x) = \sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+x}}}} $ around $0$?

It seems to me that $$e^x=1+\frac{1}{\sqrt{\pi }}{\int_0^x \frac{e^t \text{erf}\left(\sqrt{t}\right)}{\sqrt{x-t}} \, dt}$$ This integral seems to converge for all $x\in\mathbb{C}$

I came upon this conjecture by following the instructions here to do a half integral twice. Can anyone prove this conjecture is true?

- solve $\sin(z)=-1$ in the set of complex numbers
- If $U$ is connected, any two sections $U \to \mathfrak S$ either coincide or have disjoint images (Is my proof correct?)
- Help calculating $\int_C e^{-1/z}\sin(1/z)dz$ over the unit circle?
- Analytical continuation of moment generating function
- Uniform convergence of Taylor series
- Detailed proof of why integral over the upper semi-circle in $C$ of $\frac{e^{ix}}{x^2 + a^2}$ goes to $0$ as the radius goes to $\infty$?

- For fixed $z_i$s inside the unit disc, can we always choose $a_i$s such that $\left|\sum_{i=1}^n a_iz_i\right|<\sqrt3$?
- What is the value of $\int_{-\infty}^{\infty} \frac{e^{-ix}}{x^{2 }+ 4} dx$
- Complex integral
- Convergence of infinite product $\prod_{n=2}^\infty (1- \frac 1n) $
- Complex polynomial of degree $n$
- Vanishing of Taylor series coefficient
- Prove Laurent Series Expansion is Unique
- Intersections of the level curves of two (conjugate) harmonic functions
- Evaluation of $\int_0^1 \frac{\log^2(1+x)}{x} \ dx$
- Finding the correct contour to evaluate an integral with finite bounds of integration

The identity

$$

1=e^{-x}+\frac{1}{\sqrt{\pi }}{\int_0^x \frac{e^{-(x-t)} }{\sqrt{x-t}}\cdot \text{erf}\left(\sqrt{t}\right) \, dt}

$$

is true because of Laplace transforms

$$

\mathcal{L}\left(1\right)=\frac{1}{p},\qquad

\mathcal{L}\left(e^{-t}\right)=\frac{1}{p+1},

$$

$$

\mathcal{L}\left(\frac{e^{-t} }{\sqrt{t}}\right)=\int_0^\infty\frac{e^{-t} }{\sqrt{t}}e^{-pt}\ dt=\frac{1}{\sqrt{p+1}}\int_0^\infty \frac{e^{-t}}{\sqrt{t}}dt=\frac{\sqrt{\pi}}{\sqrt{p+1}},

$$

$$

\mathcal{L}\left( \text{erf}(\sqrt{t}) \right)=\int_0^\infty e^{-pt}\ dt\frac{2}{\sqrt{\pi}}\int_0^\sqrt{t}e^{-x^2}dx=\\\frac{2}{\sqrt{\pi}}\int_0^\infty \frac{e^{-pt}}{p}\frac{d}{dt}\left(\int_0^\sqrt{t}e^{-x^2}dx\right) dt=\\

\frac{2}{\sqrt{\pi}}\int_0^\infty \frac{e^{-pt}}{p}\frac{e^{-t}}{2\sqrt{t}}dt=\frac{1}{\sqrt{\pi}p\sqrt{p+1}}\int_0^\infty \frac{e^{-t}}{\sqrt{t}} dt=\frac{1}{p\sqrt{p+1}}

$$

and convolution theorem for Laplace transform

$\mathcal{L}\left(\int_0^\tau f(t)g(\tau-t)dt\right)=\mathcal{L}(f(t))\cdot \mathcal{L}(g(t))$:

$$

\frac{1}{p}=\frac{1}{p+1}+\frac{1}{\sqrt{\pi }}\cdot\frac{\sqrt{\pi}}{\sqrt{p+1}}\cdot \frac{1}{p\sqrt{p+1}}

$$

- A less challenging trivia problem
- Is the inverse of a symmetric matrix also symmetric?
- Intersection of neighborhoods of 0. Subgroup?
- Defining division by zero
- Derivative of $x^x$ at $x=1$ from first principles
- Analytical continuation of complete elliptic integral of the first kind
- Element of, subset of and empty sets
- A sheaf whose stalks are zero outside a closed subset
- Examples of uncountable sets with zero Lebesgue measure
- When are minimal faithful modules over algebras unique?
- Exponential Law for based spaces
- How to apply Thompson's A×B lemma to show this nice feature of characteristic p groups?
- Number of normal subgroups of a non abelian group of order $21$… CSIR December $2013$
- Finding a closed subalgebra generated by functions.
- irreducibility of $x^{5}-2$ over $\mathbb{F}_{11}$.