Solve Difficult Integral: Analytic Approximations Welcome

  • Context: Graduate 
  • Thread starter Thread starter Rednas
  • Start date Start date
  • Tags Tags
    Integral
Click For Summary

Discussion Overview

The thread discusses a challenging integral involving trigonometric functions and square roots, seeking analytic approximations rather than numerical solutions. The integral is presented in the context of a double integral and involves parameters that may affect its complexity. Participants express varying levels of understanding and seek clarification on the notation used.

Discussion Character

  • Technical explanation, Debate/contested, Conceptual clarification

Main Points Raised

  • The original poster (OP) presents a complex integral and expresses doubt about its solvability, emphasizing the desire for an analytic solution.
  • Some participants find the notation difficult to understand and request a clearer presentation of the integral.
  • One participant suggests that the integral relates to the Fourier Integral Theorem and discusses its interpretation in terms of wave equations, proposing that it may not be as complex as it appears when transformed into different coordinate systems.
  • There is a discussion about the use of Mathematica syntax versus more readable formats, with participants acknowledging the difficulty in deciphering the OP's initial post.

Areas of Agreement / Disagreement

Participants generally agree that the integral is challenging and that clarity in notation is needed. However, there is no consensus on the integral's complexity or the best approach to solve it, with differing opinions on its nature and solvability.

Contextual Notes

Some participants mention the need for approximations under specific conditions (e.g., when y=0 and a is small), indicating that the integral's behavior may vary significantly based on these parameters.

Rednas
Messages
2
Reaction score
0
I fear that this one is really hard, if not impossible, but an analytic answer would be way more usefull than a numerical one. Who can help me in the right direction?




[itex]\int_0^{arccos(a)} d\phi \frac{cos(\phi)}{(cos(\phi)+\sqrt{cos^2(\phi) - a^2})(x \sqrt{cos^2(\phi)-a^2}+y sin(\phi) + z + x_0 cos(\phi) )}[/itex]

with 0<a<1 and the phi integral only over positive values of the squareroot

Approximations for y=0 and a small are also welcome.

This integral comes from the double integral [itex]\int_0^{\infty} dk\int_0^{2\pi}d\phi \frac{cos(\phi)}{(cos(\phi)+\sqrt{cos^2(\phi) - a^2}} e^{i k(x \sqrt{cos^2(\phi)-a^2}+y sin(\phi) + z + x_0 cos(\phi) )}[/itex]
 
Last edited:
Physics news on Phys.org
i don't understan anything of what you wrote.
could you post somthing more readible...

use the buttons if you don't understand tex code.

ciao
marco
 
Marco_84 said:
i don't understan anything of what you wrote.
could you post somthing more readible...

use the buttons if you don't understand tex code.

ciao
marco

Yeah i agree with u! i am staring at it for a few minutes but it is quite hard to dechiper, quite ambiguous!
 
sutupidmath said:
Yeah i agree with u! i am staring at it for a few minutes but it is quite hard to dechiper, quite ambiguous!
Ambiguous? I don't think that word means what you think it means. :-p The expression is written in the syntax of Mathematica. (and I'm pretty sure it's syntactically correct)
 
Hurkyl said:
Ambiguous? I don't think that word means what you think it means. :-p The expression is written in the syntax of Mathematica. (and I'm pretty sure it's syntactically correct)
Yeah i guess! I just wanted to say that it is hard to read what the OP posted, but i guess i said the wrong way!
 
Posted something more readable. Wasn't aware of the Tex possibilities at first, so I used the Mathematica syntax.
 
Rednas said:
I fear that this one is really hard, if not impossible, but an analytic answer would be way more usefull than a numerical one. Who can help me in the right direction?




[itex]\int_0^{arccos(a)} d\phi \frac{cos(\phi)}{(cos(\phi)+\sqrt{cos^2(\phi) - a^2})(x \sqrt{cos^2(\phi)-a^2}+y sin(\phi) + z + x_0 cos(\phi) )}[/itex]

with 0<a<1 and the phi integral only over positive values of the squareroot

Approximations for y=0 and a small are also welcome.

This integral comes from the double integral [itex]\int_0^{\infty} dk\int_0^{2\pi}d\phi \frac{cos(\phi)}{(cos(\phi)+\sqrt{cos^2(\phi) - a^2}} e^{i k(x \sqrt{cos^2(\phi)-a^2}+y sin(\phi) + z + x_0 cos(\phi) )}[/itex]

Let's hope someone will take the time to work this out, because it really looks nasty!
 
Probably, unfamiliarity with the advanced topic is creating the problem. But, as I know, this particular integral is of Fourier Integral Theorem, and it solved from the left to the right i.e. formal tactics of a regular double integration not necessarily followed. This means, this integral does not generates a surface of the solid, but rather simplifies the wave equation from amplitude to frequency domain. Or the same analogy holds also for any sinusoidal signal wave. The original question was not that complex, when all variables get converted into spherical or cylindrical parametric form, it looks quite messy, but in reality, it is only a tedious job, not genuine.
 

Similar threads

Undergrad Calculating the change of the volume of a sphere using this integral
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Non solvable integral? (dx/dt)^2 dt
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Help needed with derivation: solving a complex double integral
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Spatial Fourier Transform: Bessel x Sinusoidal
  • · Replies 3 ·
Replies
3
Views
2K
MHB Checking the Orientation of an Integral on a Surface Bounded by a Sphere
  • · Replies 16 ·
Replies
16
Views
3K
Graduate How do I apply the method of steepest descents to this integral?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Can You Integrate Unit Vector ρ Without Switching to Cartesian Coordinates?
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Integration Using Trigonometric Substitution
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Multiple integral Jacobian confusion
  • · Replies 7 ·
Replies
7
Views
2K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
3K