- #1
M_1
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Is it possible to express (cos([itex]\theta[/itex])sin([itex]\theta[/itex]))^2 in terms of spherical harmonics?
Trigonometric function expanded in spherical harmonics
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Discussion Overview
The discussion revolves around the possibility of expressing trigonometric functions, specifically \((\cos(\theta)\sin(\theta))^2\) and \(\frac{1}{4}(\sin(2\theta))^2\), in terms of spherical harmonics. Participants explore the requirements for such expansions, including the need for linear combinations of spherical harmonics without trigonometric functions.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants propose that \((\cos(\theta)\sin(\theta))^2\) can be expressed in terms of spherical harmonics, with one example provided involving a specific combination of spherical harmonics.
- Others question whether a similar expression can be derived for \(\frac{1}{4}(\sin(2\theta))^2\) and emphasize the need for a linear expansion solely in spherical harmonics.
- A participant expresses uncertainty about the feasibility of such expansions and seeks guidance on how to begin the process.
- One participant mentions the importance of having no \(\phi\) dependence in the context of their calculations related to vibrational energy of a droplet.
- A later reply provides a specific expansion for the integral involving \(\sin(\theta)\cos(\theta)\sin(\theta)\) in terms of spherical harmonics, suggesting a potential solution to the posed problem.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the general feasibility of expressing the trigonometric functions in terms of spherical harmonics, as some express uncertainty while others provide specific expansions. The discussion remains unresolved regarding the broader applicability of these expressions.
Contextual Notes
Participants note the requirement for linear combinations of spherical harmonics without trigonometric functions, which may limit the scope of potential solutions. Additionally, the context of the calculations involving vibrational energy introduces specific conditions that may affect the discussion.
- #2
tiny-tim
Science Advisor
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Hi M_1! 
Isn't that 1/4 sin22θ ?
M_1 said:
Isn't that 1/4 sin22θ ?
- #3
DrClaude
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Yes, but it depends on what you want. For instance,
$$
-\frac{32 \pi^2}{9} Y_1^{-1} Y_1^1 \left(Y_1^0\right)^2 = \cos^2 \theta \sin^2 \theta
$$
But if you require a result in terms of a sum of spherical harmonics, then this requires more work.
$$
-\frac{32 \pi^2}{9} Y_1^{-1} Y_1^1 \left(Y_1^0\right)^2 = \cos^2 \theta \sin^2 \theta
$$
But if you require a result in terms of a sum of spherical harmonics, then this requires more work.
- #4
M_1
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Hi!
Thanks! Yes but then the next question is if it is possible to express 1/4 (sin(2θ)))^2 in terms of spherical harmonics.
For example we can write cos2(θ)=1/3(4*[itex]\sqrt{\pi /5}[/itex]Y20+1),
but is such an expansion possible also for 1/4 (sin(2θ)))^2? I need an a linear expansion in only spherical harmonics (not combined with trigonometric functions).
Thanks! Yes but then the next question is if it is possible to express 1/4 (sin(2θ)))^2 in terms of spherical harmonics.
For example we can write cos2(θ)=1/3(4*[itex]\sqrt{\pi /5}[/itex]Y20+1),
but is such an expansion possible also for 1/4 (sin(2θ)))^2? I need an a linear expansion in only spherical harmonics (not combined with trigonometric functions).
- #5
DrClaude
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Telling us what you need this for will help us help you.M_1 said:
- #6
M_1
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Hi DrClaude,
Thanks! But as you correctly guessed I need a sum of spherical harmonics. I would be glad to do the work but I don't know how to start and more scaringly I'm not sure if it is possible. That's why I first would like to find out if it is possible.
Thanks! But as you correctly guessed I need a sum of spherical harmonics. I would be glad to do the work but I don't know how to start and more scaringly I'm not sure if it is possible. That's why I first would like to find out if it is possible.
- #7
DrClaude
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Is it important that you have no ##\phi## dependence?
- #8
M_1
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Hi DrClaude,
I'm trying do calculate the exact solution for the vibrational energy of an inviscid droplet with a predetermined surface tension, radius, and amplitude.
In order to do so I need to solve the integral
[itex]\int[/itex]r5sin([itex]\theta[/itex])cos([itex]\theta[/itex])sin([itex]\theta[/itex])d[itex]\theta[/itex]d[itex]\varphi[/itex] where
r=r0(1+[itex]\alpha[/itex]Y20sin([itex]\varpi[/itex]t))
where [itex]\alpha[/itex] is the relative amplitude.
I try do use the nice methodology of Baxansky and Kiryati
http://dx.doi.org/10.1016/j.patcog.2006.06.001
but I get stuck on this issue.
There can be no [itex]\varphi[/itex]-dependence.
I'm trying do calculate the exact solution for the vibrational energy of an inviscid droplet with a predetermined surface tension, radius, and amplitude.
In order to do so I need to solve the integral
[itex]\int[/itex]r5sin([itex]\theta[/itex])cos([itex]\theta[/itex])sin([itex]\theta[/itex])d[itex]\theta[/itex]d[itex]\varphi[/itex] where
r=r0(1+[itex]\alpha[/itex]Y20sin([itex]\varpi[/itex]t))
where [itex]\alpha[/itex] is the relative amplitude.
I try do use the nice methodology of Baxansky and Kiryati
http://dx.doi.org/10.1016/j.patcog.2006.06.001
but I get stuck on this issue.
There can be no [itex]\varphi[/itex]-dependence.
Last edited by a moderator:
- #9
DrClaude
Mentor
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I've got it. The answer is
$$
\frac{4}{15} \sqrt{\pi } Y_0^0(\theta ,\phi )+\frac{4}{21} \sqrt{\frac{\pi }{5}} Y_2^0(\theta ,\phi )-\frac{16}{105} \sqrt{\pi } Y_4^0(\theta ,\phi )
$$
You can check it for yourself in WolframAlpha. Look at the "Alternate forms".
$$
\frac{4}{15} \sqrt{\pi } Y_0^0(\theta ,\phi )+\frac{4}{21} \sqrt{\frac{\pi }{5}} Y_2^0(\theta ,\phi )-\frac{16}{105} \sqrt{\pi } Y_4^0(\theta ,\phi )
$$
You can check it for yourself in WolframAlpha. Look at the "Alternate forms".
- #10
M_1
- 30
- 1
Excellent! Problem solved. Many thanks DrClaude!
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