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Results tagged with special-functions
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user 116983
Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
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
833
views
A special solution to the Hermite Differential Equation
I know that the general form solution to the Hermite differential equation
$$ y''-2xy'+2\lambda y=0$$
is
$$y(x)=a_1 M(-\frac{\lambda}{2},\frac{1}{2},x^2)+a_2 H(\lambda,x),$$
where $M(\cdot,\cdot,\cdot …
- 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