# Finding the limit of the following functions involving the Bessel function of the first kind

I want to find the value of the functions $f[x,y,\theta]$ and $g[x,y,\theta]$ involving the Bessel function of the first kind, a Gaussian and a linear/algebraic function in the limit $\theta\rightarrow\frac{\pi}{2}$.

I am trying to work out the limits of

f[x_, y_, θ_] := (0.835817 E^(I x + I y - 1.25119 (x + y)^2)
BesselJ[0, Sqrt[-y^2 + 400Cos[θ]]Sqrt[-2 + 2 Sec[θ]]]
(2.50239 (x + y)-I Sqrt[Cos[θ]]))/Sqrt[Cos[θ]]

g[x_, y_, θ_] := (0.835817 E^(I x + I y - 1.25119 (x + y)^2) y
BesselJ[1, Sqrt[-y^2 + 400 Cos[θ]]
Sqrt[-2 + 2 Sec[θ]]] Sqrt[-2 + 2 Sec[θ]])/(Sqrt[
Cos[θ]] Sqrt[-y^2 + 400 Cos[θ]])


On using the commands,

Limit[f[x, y, θ], θ -> π/2]


and

Limit[g[x, y, θ], θ -> π/2]


Mathematica doesn't evaluate the limits. Since this is a part of a theoretical problem (in physics) I'm working on, I should be able to evaluate these limits perfectly so that it conforms to the existing (established) results otherwise it should imply that I made a mistake while arriving at the specific forms of $f[x,y,\theta]$ and $g[x,y,\theta]$.

Is there a way to compute this?

## 1 Answer

Try this:

Series[f[x, y, θ], {θ, π/2, 0}]//Normal


I do not show the result, since it is much too long.

Have fun!

• I tried this and I did get a lengthy expression. I brought it to its simplest form but still unable to evaluate the limit. Any further help? – Jayanth Jayakumar Feb 16 '17 at 15:07
• Not that I know. I think that that is, how this limit is. But within Mma you can still do everything with this expression, plot it, or calculate some specific values. If you need to do somethin more, such as manipulate with it analytically, it is anothe story. Let me know in that case. – Alexei Boulbitch Feb 16 '17 at 15:35