How Does the Wronskian Relate to Airy Functions Ai(x) and Bi(x)?

In summary: It must be the reason why it is mentioned in the wikipedia page of Airy function.In summary, we are trying to show that the Airy functions $Ai(x)$ and $Bi(x)$ satisfy the equation $W[Ai(x),Bi(x)]=1/\pi$. After attempting to compute it directly and getting stuck, we are now considering using complex integration contour. Additionally, we want to show that $Bi(x)$ and $Bi'(x)$ are always positive for $x>0$ and prove the asymptotic identities for $Bi(x)$ and $Bi'(x)$. One approach to finding the asymptotic expansion is to use the fact that $Ai(x)$ and $Bi(x)$ are solutions to $y''
  • #1
Alone
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I am trying to show that that the Airy functions defined below satisfy: $W[Ai(x),Bi(x)]=1/\pi$.

$$Ai(x)=\frac{1}{\pi} \int_0^\infty \cos(t^3/3+xt)dt$$

$$Bi(x)=\frac{1}{\pi}\int_0^\infty \bigg[ \exp(-t^3/3+xt)+\sin(t^3/3+xt)\bigg]dt $$

I tried to compute it directly but I got stuck, here's the last term I got:

$$Ai(x)Bi'(x)-Ai'(x)Bi(x) = \frac{1}{\pi^2}\bigg[ \int_0^\infty \cos(t^3/3+xt)dt \int_0^\infty \bigg( s\exp(-s^3/3+xs)+s\cos(s^3/3+xs)\bigg) ds + \int_0^\infty \sin(t^3/3+xt)tdt\int_0^\infty \bigg(\exp(-s^3/3+xs)+\sin(s^3/3+xs)\bigg)ds \bigg]$$

I don't see how to proceed from here, I guess I need complex integration contour but how exactly?

Thanks.
I want also to show that $Bi(x),Bi'(x)>0 \forall x>0$, and to conclude the asymptotic identities:
$$Bi(x) \sim \pi^{-1/2}x^{-1/4}\exp(2/3 x^{3/2})$$

$$Bi'(x)\sim \pi^{-1/2}x^{1/4}\exp(2/3 x^{3/2})$$
 
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  • #2
I've asked my question also in MSE, it seems the Wronskian question is answered, I still will appreciate if someone were to show me how to show the asymptotic identities.
 
  • #3
Hi Alan,

One idea that would get you there is to use the fact that $Ai(x)$ and $Bi(x)$ are solutions to $y''-xy=0.$ Compute $W'[Ai(x),Bi(x)]$ and use $y''-xy=0$ to obtain that $W[Ai(x),Bi(x)]$ is a constant. To show that $W[Ai(x),Bi(x)=1/\pi,$ use the values of $Ai(x), Ai'(x), Bi(x),$ and $Bi'(x)$ at zero (see https://en.wikipedia.org/wiki/Airy_function) and the duplication formula for the gamma function (see https://en.wikipedia.org/wiki/Gamma_function)
 
  • #4
Alan said:
I still will appreciate if someone were to show me how to show the asymptotic identities.
There is a detailed discussion of how to find the asymptotic expansion of $\operatorname{Bi}(x)$ http://math.arizona.edu/~meissen/docs/asymptotics.pdf (but be warned that it takes 16 pages). Presumably you get the formula for $\operatorname{Bi}'(x)$ by differentiating the one for $\operatorname{Bi}(x)$.
 
  • #5
Opalg said:
There is a detailed discussion of how to find the asymptotic expansion of $\operatorname{Bi}(x)$ http://math.arizona.edu/~meissen/docs/asymptotics.pdf (but be warned that it takes 16 pages). Presumably you get the formula for $\operatorname{Bi}'(x)$ by differentiating the one for $\operatorname{Bi}(x)$.

I am used of reading quite a lot, that's maths and physics for you... :-)
 
  • #6
GJA said:
Hi Alan,

One idea that would get you there is to use the fact that $Ai(x)$ and $Bi(x)$ are solutions to $y''-xy=0.$ Compute $W'[Ai(x),Bi(x)]$ and use $y''-xy=0$ to obtain that $W[Ai(x),Bi(x)]$ is a constant. To show that $W[Ai(x),Bi(x)=1/\pi,$ use the values of $Ai(x), Ai'(x), Bi(x),$ and $Bi'(x)$ at zero (see https://en.wikipedia.org/wiki/Airy_function) and the duplication formula for the gamma function (see https://en.wikipedia.org/wiki/Gamma_function)

I think Euler's reflection formula for Gamma function solves this immediately.
 

FAQ: How Does the Wronskian Relate to Airy Functions Ai(x) and Bi(x)?

What is the Wronskian of Airy functions?

The Wronskian of Airy functions is a mathematical concept used to determine the linear independence of two functions. It is defined as the determinant of a matrix formed by the derivatives of the two Airy functions.

How is the Wronskian of Airy functions calculated?

The Wronskian of Airy functions can be calculated by taking the first and second derivatives of the two Airy functions and forming a 2x2 matrix. The determinant of this matrix is then taken to obtain the Wronskian.

What is the significance of the Wronskian of Airy functions?

The Wronskian of Airy functions plays an important role in the study of differential equations and the theory of linear independence. It is also used in applications such as quantum mechanics and fluid mechanics.

Can the Wronskian of Airy functions be zero?

Yes, the Wronskian of Airy functions can be zero in certain cases. This indicates that the two Airy functions are linearly dependent and do not form a fundamental set of solutions.

Are there any relationships between the Wronskian of Airy functions and other mathematical concepts?

Yes, the Wronskian of Airy functions has connections to other mathematical concepts such as the Airy differential equation, Bessel functions, and Sturm-Liouville theory. It is also related to the concept of linear independence in linear algebra.

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