Harmonic oscillator modeled with a contour integral

[Ingrid]

Homework Statement

Greetings, gents.

I have a modelization problem you might be able to help me with...

I have two oscillators, modeled as:
$$osc_{1}=\cos{(a z)}$$$$osc_{2}=\cos{(\frac{b}{z})}$$
and a resonance condition f(z) when these two oscillators are combined, modeled as:
$$f(z)=\frac{1}{2-\cos{(a z)}-\cos{(\frac{b}{z})}}$$

Assume, for illustration purposes, that $$a=10\pi$$ and $$b=70\pi$$
Then, if z=5, we have:
$$f(5)=\frac{1}{2-\cos{(10\pi 5)}-\cos{(\frac{70\pi}{5})}}=\frac{1}{2-1-1}=\frac{1}{0}=\infty$$
That is, the function f(z) would have a pole at z=5 (in addition to the trivial poles at z=1 and z=2)

I'm trying to find a quick way to check whether a value z, for a given value of a and b (not necessarily 10 π
and 70 π), is inferior to the resonance value at which f(z) becomes infinite (at z=5 in the example above).

f(z) can be seen as a complex-valued function of a complex variable z. Looking at the complex plane from
above, we'd have three poles on the real axis, at 1, 2 and 5, indicated with a "P" in the picture below:

       ^
       | i axis
       |
       |
       |
-------*---P---P---.---.---P---.---.---.--->
      0|   1   2   3   4   5   6   7 . .   Real axis
       |
       |

The value of the function at z=0 is complicated by the factor cos(b/z) but doesn't interest us for now.

I thought about inspiring myself from the residue theorem of complex analysis.
Assume I'm integrating f(z) along the contour of a circle "C₁" of radius 1, centered on 3.5+0i.
That contour won't contain any poles, and, from the Cauchy Integral theorem, the value of that
contour integral should thus be zero.
$$\oint_{C_1} \frac{1}{2-\cos{(10\pi 5)}-\cos{(\frac{70\pi}{5})}}\,dz = 0$$

Now, if I were to integrate f(z) along a larger contour, say, a circle "C₂" of radius 2, centered
on 4.5+0i, that contour would encompass exactly one pole, the one at 5+0i.
The value of the contour integral should then be the residue at 5+0i.
$$\oint_{C_2} \frac{1}{2-\cos{(10\pi 5)}-\cos{(\frac{70\pi}{5})}}\,dz = \operatorname{Res}( f, 5+0i )$$

Note that the circular contours I'm interested in are always centered on a point of the real axis.
Furthermore, the circular contour's radius, as well as its center's position, are chosen so that the
poles with real parts less than or equal to 2 are not included in the contour.

I understand that the relation between the value of the contour integral and the residue at the pole
included in the contour would depend on the nature of the singularity at 5+0i — e.g. removable,
or multiple, or essential.

A quick value check shows that, with a=10 π and b=70 π:
$$(z-5)f(z)=\infty$$
whilst
$$(z-5)(z-5)f(z) = 0$$
We thus seem to have a second-order pole at 5+0i.The attempt at a solution

Where I'm stumbling is the calculation of a closed analytical form of the following contour integral
along a circle centered on the real axis:
$$\oint_{C} \frac{1}{2-\cos{(a z)}-\cos{(\frac{b}{z})}}\,dz$$
I've tried the usual integration tricks:

• subsituting cos(z) with (e^iz+e^-iz)/2
• parametrizing the circular contour integral as an integration over a single variable t in [0..2π] using the z=e^it substitution

to no avail.

Note that the derivative of f(z) is:
$$\frac{df}{dz}=\frac{b\sin{(\frac{b}{z})}-a z^2 \sin{(a z)}}{z^2(2-\cos{(a z)}-\cos{(\frac{b}{z})})^2}$$
Given that the derivative's denominator becomes zero at the same points as the denominator of f(z), the locations
of the derivative's poles are the same as those of f(z).

So, I think that either the value of the contour integral of f(z), or of the contour integral of its derivative
$$\oint_{C} \frac{b\sin{(\frac{b}{z})}-a z^2 \sin{(a z)}}{z^2(2-\cos{(a z)}-\cos{(\frac{b}{z})})^2}\,dz$$
along a circle C of my choosing, with its center on the real axis, should provide me with a quick
and reliable indication as to whether a pole is — or isn't — included in the contour.

• Note that in the general problem that interests me, that is, for arbitrary values of a and b, the location of
  the non-trivial pole (5+0i in the illustration above) is, unfortunately, totally unknown.
• Thus, the residue cannot be calculated, as the point where it should be calculated (the pole) is unknown.
• Thus, we cannot rely on the residue theorem. The only way to calculate the contour integral — whether of f(z) or of
  its derivative — would seem to be by using fairly advanced integration techniques, and this is where I'm stumbling.

Any help on obtaining closed forms for the contour integrals of either f(z) or of its derivative
would be much appreciated !

Of course, indications as to why the evaluation of a contour integral as a quick check for the
presence of a pole within a circular contour of a given radius might work or not — e.g. at essential
singularities — would also be appreciated…

 

[kurt.physics]

I have been trying to find a solution for this problem for some time now, and I'm starting to feel a bit stuck. Any help would be greatly appreciated!