Questions tagged [airy-functions]
For questions about Airy functions, the solution to Schrödinger's equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field.
77 questions
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An Airy Integral
I am trying to evaluate $$\int_0^{\infty} \frac{1}{\sqrt{t}} \cos\left(\frac{t^3}{3} + \eta t \pm \frac{\pi}{4} \right) \, dt$$
where $\eta$ is some constant.
The context of this problem comes from ...
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Asymptotic expansion of Airy function for positive $x$ values
I am trying to derive the asymptotic expansion of the Airy function $$\operatorname{Ai}(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(xt+t^3/3)}dt$$ using the saddle point method. I deformed the ...
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The estimate of $\left\|yf'\right\|_{L^2(\mathbb{R}_+)}$ in inhomogeneous Airy equation
For the inhomogeneous airy equation $yf(y) - f''(y) = h(y)$ with boundary condition $f'\big|_{y=0}=0$. There is a solution as follows.
$$
f(y) = \sqrt{3}\pi Ai(y)\int_0^{\infty}Ai(z)h(z)dz
+...
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Solving ODEs with power series method Airy's equation
Once you've found the recurrence relation relating say for example $$ a_{k+2}=\frac{a_{k-1}}{(k+2)(k+1)}$$ in Airy's equation, and after finding pattern for $a_{3n}$ and $a_{3n+1}$, i.e $$ a_{3n} = \...
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A solution procedure by "Fourier-Laplace transform" to Airy function (from oscillatory to exponential behavior)?
In studying the Airy equations (with final objective is to find the asymptotic behavior of the oscillatory solution to growing (frequency $\omega\to\infty$) decaying ($\omega\to0$) exponential ...
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Hilbert transform of the integral of a function
Given that f and g are Hilbert transform pair
$$Hf(x) = g(x)$$
Although the derivative will maintain the transform pair relation
$$Hf'(x) = g'(x)$$
Does the Hilbert transform pair relation apply to ...
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Convergence of the integral definition of the Airy function
The airy function $\mathrm{Ai}(x)$ can be defined by an improper integral:
$$ \mathrm{Ai}(x) \equiv \frac{1}{\pi} \int^\infty_0\mathrm{d}t \cos\left(\frac{t^3}{3}+tx\right) \, . $$
I want to prove the ...
- 469
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Asymptotic expansion of $\int_{-\infty}^{\infty} e^{ix\left(\frac{t^3}{3}+t\right)}dt $
How do you find the asymptotic expansion of $\int_{-\infty}^{\infty} e^{ix\left(\frac{t^3}{3}+t\right)}dt $ as $x \to \infty$ ?
I know that I will eventually have to use the fact that $\int_{-\infty}^...
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Integration of a product of Airy function and gaussian function
I have been stuck with an integration of the following type
$$\int\frac{2^{1/3}}{x^{1/3}}\operatorname{Ai}\left(\frac{2^{1/3}(x-y)}{x^{1/3}}\right)e^{-b\left(k+x\omega\right)^2}\,dx$$
I am not so sure ...
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Limit of Ai$(x)$ from above [closed]
I wonder how to show that
$$
\text{Ai}(x)>0\text{ for }x=x_{1}+\epsilon,\epsilon>0,
$$
where $x_{1}$ is the first real zero (nearest to $x=0$) of Ai$(x)$. I want to know which formula for Ai$(x)$...
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Reducing Airy differential equation to Bessel equation
I have to reduce the following differential equation to the Bessel equation.
$$y''+\lambda xy = 0$$
Upon seeing articles on the internet, I found that this is an Airy differential equation, which can ...
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Inverse and extension of the $n$th Airy Ai and Bi zero
$\def\ai{\operatorname{ai}} \def\bi{\operatorname{bi}} \def\Ai{\operatorname{Ai}} \def\Bi{\operatorname{Bi}} $ Airy Ai Zero $\ai_n$ gives the $n$th zero of the Airy Ai function and Airy Bi Zero $\bi_n$...
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Solving the first-order non-linear differential equation $y' = y^2 - 2 x$
I am trying to solve this Cauchy's problem:
$$
y' = y^2 - 2x
$$
with condition $y(0) = 2$
It's very similar to Bernoulli equation
$$
y' + a(x)y = b(x)y^2$$
however doesn't contain $a(x)y$. I also ...
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Fourier transform of Airy function
I am looking for the Fourier transform of the Airy function $\left(\dfrac{2J_1(x)}{x}\right)^2$ where $J_1$ is the Bessel function of the first kind of order one.
Thank you.
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where does the airy function integral representation come from
I am trying to understand the WKB approximation as it appears in Griffiths QM, but it requires solving the Airy equation:
$$\frac{d^2\Psi }{dz^2}=kz\Psi$$
Solving this by using the power series method ...
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