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From wikipedia:

The Bessel function of the first kind is an entire function if α is an integer, otherwise it is a multivalued function with singularity at zero.

I have plotted the function $J_\alpha(x)$ for a few values of $\alpha$ in the $-10\le x\le 10$ interval:

Plot[{Re[BesselJ[0, x]], Im[BesselJ[0, x]]}, {x, -10, 10}]

enter image description here$\alpha = 0$

Plot[{Re[BesselJ[1, x]], Im[BesselJ[1, x]]}, {x, -10, 10}]

enter image description here$\alpha = 1$

Plot[{Re[BesselJ[-2, x]], Im[BesselJ[-2, x]]}, {x, -10, 10}] 

enter image description here$\alpha = -2$

Plot[{Re[BesselJ[5/4, x]], Im[BesselJ[5/4, x]]}, {x, -20, 20}]

enter image description here$\alpha = \frac{5}{4}$

Plot[{Re[BesselJ[-2/3, x]], Im[BesselJ[-2/3, x]]}, {x, -20, 20}]  

enter image description here$\alpha = \frac{-2}{3}$

Plot[{Re[BesselJ[Sqrt[2], x]], Im[BesselJ[Sqrt[2], x]]}, {x, -20, 20}] 

enter image description here$\alpha = \sqrt{2}$

Plot[{Re[BesselJ[-Sqrt[3], x]], Im[BesselJ[-Sqrt[3], x]]}, {x, -20, 
  20}] 

enter image description here$\alpha = -\sqrt{3}$

Plot[{Re[BesselJ[2 + I, x]], Im[BesselJ[2 + I, x]]}, {x, -20, 20}] 

enter image description here$\alpha = 2+i$

Plot[{Re[BesselJ[-2 + I, x]], Im[BesselJ[-2 + I, x]]}, {x, -20, 20}]

enter image description here$\alpha = -2+i$

Plot[{Re[BesselJ[10, x]], Im[BesselJ[10, x]]}, {x, -100, 100}]

enter image description here$\alpha = 10$

Seems that for integer values of $\alpha$, the function $J_\alpha(x)$ is real-valued but for other values $\alpha\in(\mathbb R-\mathbb Z)$ the function has complex values.

Could you please give an easy and intuitive explanation for the concepts entire function and multivalued function based on these plots?

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  • $\begingroup$ you should like put bounty on question $\endgroup$
    – user153330
    Dec 20, 2015 at 14:58

1 Answer 1

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What is going on here is that a Bessel function $B$ is defined implicitly as a solution to a differential equation. This means that if you know one value of $B$, say $y = B(z)$ (and suitable derivatives $\mathrm{d}B/\mathrm{d}z$ etc) then you can find the values of $B(z+t)$ where $z+t$ is near $z$.

In particular you want $B$ to be continuously defined. So if you let $z$ wander around the complex plane then $B(z)$ needs to vary continuously as you go.

"Entire function" means "function defined and holomorphic on the whole complex plane." That is, for every complex number $z$ the entire function $B$ has a single value $B(z)$. (And the values are nicely arranged so the function has a complex derivative: the geometric definition is that the mapping preserves angles locally).

In this case life is simple, and the function is continuous as you require.

On the other hand, you might find that as $z$ wanders there are paths where a continuous selection of $B(z)$ doesn't work: as $z$ comes back to where it started your continuous selection of $B$ has ended up somewhere different.

The easiest example of this is the square root. In polar coordinates, $\sqrt (re^{i \theta})$ = $\sqrt({r}) \, e^{i \theta / 2}$. If you let $z$ start at $1$ (with $\sqrt{z} = 1$) and move $2\pi$ around the circle of radius 1, then you get back to $$\sqrt{1} e^{i 2\pi / 2} = e^{i \pi} = -1 $$

By following a path in a circle you have come back to the other square root!

In this case there is no entire function that can define "$\sqrt{z}$" on the whole plane. But for each $z$ there are only finitely many (two!) possible square roots (call them $a$ and $b$). So if you are happy to allow $\sqrt: z \mapsto \{a, b\}$, then this is a function to a "multiple" complex plane whose range is a set of pairs of complex numbers.

We have defined a multifunction. If we start at any $z$ and one value of $\sqrt (z)$ then we can take a continuous selection of values for the square root, as an ordinary function, provided that we don't stray too far before coming back.

Except that in this case $0$ is special because it only has one value. And there's no way of taking a small neighbourhood of $0$ with $\sqrt$ continuously defined on it. But we allow ourselves such singularities as long as they are isolated.

(Similarly a multifunction can have infinitely many values as long as there are only countably many "sheets": $\log$ is like this)

These topics are covered in first-semester complex analysis courses: any introductory university text ought to be be helpful.

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