# How to evaluate the integral of exponentials

I want to evaluate the following integral wrt $\phi$ from 0 to $2 \pi$:

Integrate[
Exp[(Γ^2 (Cos[ϕ]^2 Sin[θ]^2 σx^2 + Sin[ϕ]^2 Sin[θ]^2 σy^2)
- 2 c Γ (x0 Cos[ϕ] Sin[θ] + y0 Sin[ϕ] Sin[θ]))/(2 c^2)],
{ϕ, 0, 2 π}
]


Currently Mathematica returns the input without evaluating anything. I have tried to Simplify the expression before evaluating the integral as well.

This can most likely be done by instead using NIntegrate and by using specific values for each of the parameters. However, I am really interested in obtaining the analytical result of this evaluation. Is anyone aware of a method of forcing Mathematica to generate a symbolic output for this integral.

I will also post this question on Mathematics Stack Exchange to try to source a substitution or equivalent method to break this problem down.

• "forcing Mathematica to generate a sybolic output for this integral": there are plenty of cases in which an analytical form of the integral is not known or doesn't exist. Do you have any reason to believe that yours is not one of those cases? – MarcoB Feb 12 '16 at 17:42
• First thing I'd do is look at some special cases, eg Integrate[ Exp[ 2 Sin[\[Phi]]^2 - Cos[\[Phi]] ], {\[Phi], 0, 2 Pi}]. Even this relatively simple form does not evaluate... – george2079 Feb 12 '16 at 17:59

You can get an approximate answer by doing a series expansion of the integrand. As always, the more terms used in the expansion, the better the approximation.

expr = Exp[(Γ^2 (Cos[ϕ]^2 Sin[θ]^2 σx^2 +
Sin[ϕ]^2 Sin[θ]^2 σy^2) -
2 c Γ (x0 Cos[ϕ] Sin[θ] +
y0 Sin[ϕ] Sin[θ]))/(2 c^2)];

f[n_Integer, ϕ0_] :=
Integrate[Series[expr, {ϕ, ϕ0, n}] // Normal, {ϕ, 0, 2 Pi}]


Expanding about the middle of the integration interval,

f[6, Pi] // Simplify

(*  (1/2520)E^((Γ Sin[θ] (2 c x0 + Γ \
σx^2 Sin[θ]))/(
2 c^2)) π (5040 + (
840 π^2 Γ Sin[θ] (-c x0 + Γ (y0^2 - \
σx^2 + σy^2) Sin[θ]))/c^2 - (1/(
c^6))π^6 Γ Sin[θ] (c^5 x0 +
c^4 Γ (15 x0^2 -
16 (y0^2 - σx^2 + σy^2)) Sin[θ] +
15 c^3 x0 Γ^2 (x0^2 -
5 (y0^2 - σx^2 + σy^2)) Sin[θ]^2 -
5 c^2 Γ^3 (9 x0^2 (y0^2 - σx^2 + σy^2) -
4 (y0^4 - 6 y0^2 (σx^2 - σy^2) +
3 (σx^2 - σy^2)^2)) Sin[θ]^3 +
15 c x0 Γ^4 (y0^4 - 6 y0^2 (σx^2 - σy^2) +
3 (σx^2 - σy^2)^2) Sin[θ]^4 - Γ^5 \
(y0^6 - 15 y0^4 (σx^2 - σy^2) +
45 y0^2 (σx^2 - σy^2)^2 -
15 (σx^2 - σy^2)^3) Sin[θ]^5) + (1/(c^4))
42 π^4 Γ Sin[θ] (c^3 x0 + Γ Sin[\
θ] (c^2 (3 x0^2 -
4 (y0^2 - σx^2 + σy^2)) + Γ Sin[\
θ] (-6 c x0 (y0^2 - σx^2 + σy^2) + Γ (y0^4 \
+ 3 (σx^2 - σy^2)^2 +
6 y0^2 (-σx^2 + σy^2)) Sin[θ]))))  *)

• could you explain the series expansion of the integrand about the middle of the integration limits $\pi$? How is this a valid expansion? – Sid Feb 13 '16 at 22:00
• @Sid - If you need help with using the Command Series see the linked documentation page. If you need help understanding the concept of Taylor series see any online documentation such as the linked wiki page. – Bob Hanlon Feb 13 '16 at 22:13