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--- math121b:03-30 [2020/03/30 15:43]
pzhou [Speherical Bessel function $j_n(x), y_n(x)$]
+++ math121b:03-30 [2026/02/21 14:41] (current)
@@ Line 15: / Line 15: @@
 What are they good for? Well, we will see the usual Bessel function is good for solving PDE in cylindrical coordinate in 3D; these will be useful when using spherical coordinate $r, \theta, \phi$.
+==== Hankel Function ====
+$H_n^1(x), H^2_n(x)$ to $J_n(x), Y_n(x)$  are like $e^{ix}$ and $e^{-ix}$ to $\sin x, \cos x$. They are complex valued functions.
+In real life, I have encountered them when solving Dirac equation on expanding universe.
+The function is named after a German mathematician Hermann Hankel. He is also known for 'Hankel contour',some contour integral expression for $J_n$ and $H_n$
+==== Hyperbolic Bessel Function ====
+The $I_p(x)$ and $K_p(x)$ are related to Bessel function when you replace $x$ by $ix$ in the input.
+Just convenient names.
+==== Airy Function ====
+This function is pretty popular and useful. It is worth studying this in more details $Ai(x)$.
+It solves equation of the type
+$$ (d/dx)^2 y(x) - x y(x) = 0. $$
+Its solution has the property that, it is osillatory for $x < 0$ and have exponential decay for $x > 0$, indeed, the oscillation freqency is $\omega = \sqrt{-x}$, if you compare this with Harmonic oscillator
+$$ (d/dx)^2  y(x) + \omega^2 y(x) = 0$$
+The solution to which is $e^{\pm i \omega x}$ and we know imaginary $\omega$ means exponetial dampling or growth.
+The Airy function is used to model transition behavior in quantum mechanics, when you go from the 'allowed region' (total energy > potential energy) to 'forbidden region' (other wise).
+We can see the asymptotic behavior of $Ai(x)$ for $x \to -\infty$ and $x \to +\infty$,
+===== Other Special Function (Boas) =====

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