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Results tagged with real-analysis
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user 116983
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
12
votes
1
answer
351
views
A problem involving the Error Function
I am looking at the following function on the domain $x\geq 0$:
$$F(x)=(x+a)e^{x^2}(1-\mathrm{erf}(x))-\frac{b}{\sqrt\pi},$$
where $a>0$, $0<b<1$ are parameters. From plotting this function for diff …
- 389
2
votes
1
answer
191
views
A first order ODE problem
Fix $a>0$ and $b>0$. Does the following ODE
\begin{equation}
G(x)^2+2axG(x)G'(x)+2aG'(x)(x-b)=0 \tag{*}
\end{equation}
have a solution, say, $F(x)$, that satisfies $F(x)>0$ and $F'(x)<0$ on $(b,\inft …
- 389
1
vote
1
answer
170
views
Root problem involving error function
I ran into this problem in my research:
Let $y_0$ be the root of
$$-(y+a)e^{y^2}\mathit{erfc}(y)+\frac{b}{\sqrt{\pi}}=0$$
on interval $[-a,\infty)$, while $a>0$ and $0<b<1$.
How can I show
$$y_0\ …
- 389
1
vote
0
answers
35
views
Prove monotonicity of a system of difference equations
I have the following system of difference equations. Fix any $(\alpha,K,p)\in\mathbb R_+\times\mathbb R_+\times (0,1)$. Let $u_0=1, \nu_0=0$. The sequence $(a_t,w_t,\nu_t,u_t)_{t\geq 1}$ is defined as …
- 389