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I am trying to understand the difference between Bessel functions and modified Bessel functions (simply googling is yielding complicated, non-intuitive answers). I was under the impression that one allowed for a complex parameter while the other did not - is this true?

My question stems from trying to understand the radial part of the Hydrogen eigenproblem (with $u = rR(r)$):

$$ \frac{d^2u}{dr^2} = \left[ \frac{l(l+1)}{r^2} - k\right] u(r) $$

which is solved by a linear combination of Spherical Bessel functions and Neumann functions:

$$ u(r) = Ar j_l(kr) + Brn_l(kr) $$

Is this solution valid for both real and imaginary $k$?

For reference, this linear combination is from Griffiths' Introduction to Quantum Mechanics, Equation 4.45.

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up vote 5 down vote accepted

The ordinary Bessel functions are perfectly well defined for complex arguments. For example, here is a plot of $\Re[J_2(x + i y)]$:

enter image description here

The difference between the ordinary and modified Bessel functions is that they satisfy different equations:

$$ z^2 y'' + z y' + (z^2 - n^2) y = 0, $$

for the ordinary Bessel functions and

$$ z^2 y'' + z y' - (z^2 + n^2) y = 0, $$

for the modified Bessel functions.

Note that there is a relationship between them:

$$ J_{\nu }(z)=\frac{z^{\nu } I_{\nu }(i z)}{(i z)^{\nu }} $$

with similar identities going the other way. It's all very similar to the relationship between the trig functions $\sin(z),\cos(z)$ with the hyperbolic functions $\sinh(z),\cosh(z)$.

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If you want to know what the Bessel function really is, imagine 360 people standing on a circle 100+ meters in diameter and singing the same note lamda < 1 meter. Near the center of the circle there will be 100 plane waves convergin in phase, and the radial amplitude function will be the zero-order Bessel function J_0. If you want to see J_1, have every singer delay the phase of his note successively by one degree as you move around the circle, so that after going around the whole circle you are back in phase again: the radial amplitude will be J_1 along the line of the first singer, and zero along the line connecting singer 90 with singer 270. Similarly, delay the phase 2 degrees per singer and you get J_2 with two nodal lines, etc.

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If you are planning to continue down this road, I would advise you to get a copy of the book Arfken: Mathematical Methods for Physicists. There (chapter 14.5, 7th edition) the difference is explained in detail.

You use the modified Bessel Equation and their solutions (the modified Bessel functions) when you are working in cylindrical coordinates. In order to get to the modified Bessel equation, you separate the Laplace (or Helmholtz) equation(s) in cylindrical coordinates and arrive (at some point) at the sign change mentioned by Michael.

Now this sign change is pretty important, because it changes the behaviour of the solutions to the modified Bessel equation (as in comparison to the solutions to the (unmodified) Bessel equation). The solutions to the modified Bessel equation (i.e. the modified Bessel functions) are NOT oscillatory. And they show exponential behaviour.

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