Re: Extract coefficients of a trig polynomial

• To: mathgroup at smc.vnet.net
• Subject: [mg126136] Re: Extract coefficients of a trig polynomial
• From: danl at wolfram.com
• Date: Fri, 20 Apr 2012 07:46:21 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jmoge7\$46g\$1@smc.vnet.net>

```On Thursday, April 19, 2012 2:55:51 AM UTC-5, Sam Takoy wrote:
> Hi,
>
> Suppose I have an expression that is a trigonometric polynomial in
> theta. Is there a way to neatly pick out the coefficients of the
> polynomial. I find that FourierCoefficient takes quite a bit of time,
> probably because it does a lot of integrations. My coefficients are
> very complicated expressions but do not depend on theta.
>
>
> Sam
>
> PS: expr = (1/1536)(1536 BesselJ[0,\[Rho]]-72 \[Epsilon]^2 \[Rho]^2
> BesselJ[0,\[Rho]]-80 \[Epsilon]^3 \[Rho]^2 BesselJ[0,\[Rho]]-384 \
> [Epsilon] \[Rho] BesselJ[1,\[Rho]]-144 \[Epsilon]^2 \[Rho] BesselJ[1,\
> [Rho]]-80 \[Epsilon]^3 \[Rho] BesselJ[1,\[Rho]]+10 \[Epsilon]^3 \
> [Rho]^3 BesselJ[1,\[Rho]]-96 \[Epsilon]^2 \[Rho]^2 BesselJ[0,\[Rho]]
> Cos[2 \[Theta]]-120 \[Epsilon]^3 \[Rho]^2 BesselJ[0,\[Rho]] Cos[2 \
> [Theta]]-384 \[Epsilon] \[Rho] BesselJ[1,\[Rho]] Cos[2 \[Theta]]-192 \
> [Epsilon]^2 \[Rho] BesselJ[1,\[Rho]] Cos[2 \[Theta]]-120 \[Epsilon]^3 \
> [Rho] BesselJ[1,\[Rho]] Cos[2 \[Theta]]+15 \[Epsilon]^3 \[Rho]^3
> BesselJ[1,\[Rho]] Cos[2 \[Theta]]-24 \[Epsilon]^2 \[Rho]^2 BesselJ[0,\
> [Rho]] Cos[4 \[Theta]]-48 \[Epsilon]^3 \[Rho]^2 BesselJ[0,\[Rho]]
> Cos[4 \[Theta]]-48 \[Epsilon]^2 \[Rho] BesselJ[1,\[Rho]] Cos[4 \
> [Theta]]-48 \[Epsilon]^3 \[Rho] BesselJ[1,\[Rho]] Cos[4 \[Theta]]+6 \
> [Epsilon]^3 \[Rho]^3 BesselJ[1,\[Rho]] Cos[4 \[Theta]]-8 \[Epsilon]^3 \
> [Rho]^2 BesselJ[0,\[Rho]] Cos[6 \[Theta]]-8 \[Epsilon]^3 \[Rho]
> BesselJ[1,\[Rho]] Cos[6 \[Theta]]+\[Epsilon]^3 \[Rho]^3 BesselJ[1,\
> [Rho]] Cos[6 \[Theta]])

Not sure, but possibly this will do what you have in mind.

In[206]:= trigs = Union[Cases[expr, Cos[(_.)*\[Theta]], Infinity]]

Out[206]= {Cos[2*\[Theta]], Cos[4*\[Theta]], Cos[6*\[Theta]]}

In[207]:= (Coefficient[expr, #1]*#1 & ) /@ trigs

Out[207]= {(1/
1536)*((-96*\[Epsilon]^2*\[Rho]^2*BesselJ[0, \[Rho]] -
120*\[Epsilon]^3*\[Rho]^2*BesselJ[0, \[Rho]] -
384*\[Epsilon]*\[Rho]*BesselJ[1, \[Rho]] -
192*\[Epsilon]^2*\[Rho]*BesselJ[1, \[Rho]] -
120*\[Epsilon]^3*\[Rho]*BesselJ[1, \[Rho]] +
15*\[Epsilon]^3*\[Rho]^3*BesselJ[1, \[Rho]])*
Cos[2*\[Theta]]),
(1/1536)*((-24*\[Epsilon]^2*\[Rho]^2*BesselJ[0, \[Rho]] -
48*\[Epsilon]^3*\[Rho]^2*BesselJ[0, \[Rho]] -
48*\[Epsilon]^2*\[Rho]*BesselJ[1, \[Rho]] -
48*\[Epsilon]^3*\[Rho]*BesselJ[1, \[Rho]] +
6*\[Epsilon]^3*\[Rho]^3*BesselJ[1, \[Rho]])*
Cos[
4*\[Theta]]), ((-8*\[Epsilon]^3*\[Rho]^2*BesselJ[0, \[Rho]] -
8*\[Epsilon]^3*\[Rho]*BesselJ[1, \[Rho]] +
\[Epsilon]^3*\[Rho]^3*BesselJ[1, \[Rho]])*Cos[6*\[Theta]])/
1536}

Daniel Lichtblau
Wolfram Research

```

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