Re: Extract coefficients of a trig polynomial

• To: mathgroup at smc.vnet.net
• Subject: [mg126150] Re: Extract coefficients of a trig polynomial
• From: "Dr. Wolfgang Hintze" <weh at snafu.de>
• Date: Sat, 21 Apr 2012 00:29:59 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jmoge7\$46g\$1@smc.vnet.net>

```On 19 Apr., 09:55, Sam Takoy <sam.ta... at yahoo.com> 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.
>
> Many thanks in advance,
>
> 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]])
>
>

Hi,

to avoid special characters I'm gong to rewrite expr as

In[14]:=
expr = (1/1536)*(1536*BesselJ[0, r] - 72*q^2*r^2*BesselJ[0, r] -
80*q^3*r^2*BesselJ[0, r] - 384*q*r*BesselJ[1, r] -
144*q^2*r*BesselJ[1, r] - 80*q^3*r*BesselJ[1, r] +
10*q^3*r^3*BesselJ[1, r] - 96*q^2*r^2*BesselJ[0, r]*Cos[2*t] -
120*q^3*r^2*BesselJ[0, r]*Cos[2*t] - 384*q*r*BesselJ[1,
r]*Cos[2*t] - 192*q^2*r*BesselJ[1, r]*Cos[2*t] - 120*q^3*r*BesselJ[1,
r]*Cos[2*t] + 15*q^3*r^3*BesselJ[1, r]*Cos[2*t] -
24*q^2*r^2*BesselJ[0, r]*Cos[4*t] - 48*q^3*r^2*BesselJ[0, r]*Cos[4*t]
-
48*q^2*r*BesselJ[1, r]*Cos[4*t] - 48*q^3*r*BesselJ[1, r]*Cos[4*t]
+ 6*q^3*r^3*BesselJ[1, r]*Cos[4*t] - 8*q^3*r^2*BesselJ[0, r]*Cos[6*t]
- 8*q^3*r*BesselJ[1, r]*Cos[6*t] + q^3*r^3*BesselJ[1, r]*Cos[6*t])

Out[14]=
(1/1536)*(1536*BesselJ[0, r] - 72*q^2*r^2*BesselJ[0, r] -
80*q^3*r^2*BesselJ[0, r] - 384*q*r*BesselJ[1, r] -
144*q^2*r*BesselJ[1, r] - 80*q^3*r*BesselJ[1, r] +
10*q^3*r^3*BesselJ[1, r] - 96*q^2*r^2*BesselJ[0, r]*Cos[2*t] -
120*q^3*r^2*BesselJ[0, r]*Cos[2*t] - 384*q*r*BesselJ[1, r]*Cos[2*t]
- 192*q^2*r*BesselJ[1, r]*Cos[2*t] - 120*q^3*r*BesselJ[1, r]*Cos[2*t]
+ 15*q^3*r^3*BesselJ[1, r]*Cos[2*t] - 24*q^2*r^2*BesselJ[0,
r]*Cos[4*t] - 48*q^3*r^2*BesselJ[0, r]*Cos[4*t] -
48*q^2*r*BesselJ[1, r]*Cos[4*t] - 48*q^3*r*BesselJ[1, r]*Cos[4*t] +
6*q^3*r^3*BesselJ[1, r]*Cos[4*t] - 8*q^3*r^2*BesselJ[0, r]*Cos[6*t] -
8*q^3*r*BesselJ[1, r]*Cos[6*t] + q^3*r^3*BesselJ[1, r]*Cos[6*t])

> I would use Cases with apropriate patterns like this
>
> First make a list out of your expression:
>
In[15]:=
el = List @@ (List @@ expr)[[2]]

Out[15]=
{1536*BesselJ[0, r], -72*q^2*r^2*BesselJ[0, r], -80*q^3*r^2*BesselJ[0,
r], -384*q*r*BesselJ[1, r], -144*q^2*r*BesselJ[1, r],
-80*q^3*r*BesselJ[1, r], 10*q^3*r^3*BesselJ[1, r],
-96*q^2*r^2*BesselJ[0, r]*Cos[2*t], -120*q^3*r^2*BesselJ[0,
r]*Cos[2*t],
-384*q*r*BesselJ[1, r]*Cos[2*t], -192*q^2*r*BesselJ[1, r]*Cos[2*t],
-120*q^3*r*BesselJ[1, r]*Cos[2*t], 15*q^3*r^3*BesselJ[1, r]*Cos[2*t],
-24*q^2*r^2*BesselJ[0, r]*Cos[4*t], -48*q^3*r^2*BesselJ[0,
r]*Cos[4*t], -48*q^2*r*BesselJ[1, r]*Cos[4*t],
-48*q^3*r*BesselJ[1, r]*Cos[4*t], 6*q^3*r^3*BesselJ[1, r]*Cos[4*t],
-8*q^3*r^2*BesselJ[0, r]*Cos[6*t], -8*q^3*r*BesselJ[1, r]*Cos[6*t],
q^3*r^3*BesselJ[1, r]*Cos[6*t]}
>
> Then collect all terms the desired patterns in Cos[]
>
> Cos with any agrument:
>
In[22]:=
c1 = Cases[el, _*Cos[_]]

Out[22]=
{-96*q^2*r^2*BesselJ[0, r]*Cos[2*t], -120*q^3*r^2*BesselJ[0,
r]*Cos[2*t], -384*q*r*BesselJ[1, r]*Cos[2*t], -192*q^2*r*BesselJ[1,
r]*Cos[2*t], -120*q^3*r*BesselJ[1, r]*Cos[2*t], 15*q^3*r^3*BesselJ[1,
r]*Cos[2*t], -24*q^2*r^2*BesselJ[0, r]*Cos[4*t],
-48*q^3*r^2*BesselJ[0, r]*Cos[4*t], -48*q^2*r*BesselJ[1,
r]*Cos[4*t], -48*q^3*r*BesselJ[1, r]*Cos[4*t], 6*q^3*r^3*BesselJ[1,
r]*Cos[4*t], -8*q^3*r^2*BesselJ[0, r]*Cos[6*t], -8*q^3*r*BesselJ[1,
r]*Cos[6*t], q^3*r^3*BesselJ[1, r]*Cos[6*t]}
>
> Cos[2 times anything] and so on
>
In[17]:=
Cases[el, _*Cos[2*_]]

Out[17]=
{-96*q^2*r^2*BesselJ[0, r]*Cos[2*t], -120*q^3*r^2*BesselJ[0,
r]*Cos[2*t], -384*q*r*BesselJ[1, r]*Cos[2*t], -192*q^2*r*BesselJ[1,
r]*Cos[2*t], -120*q^3*r*BesselJ[1, r]*Cos[2*t], 15*q^3*r^3*BesselJ[1,
r]*Cos[2*t]}
>
In[18]:=
Cases[el, _*Cos[4*_]]

Out[18]=
{-24*q^2*r^2*BesselJ[0, r]*Cos[4*t], -48*q^3*r^2*BesselJ[0,
r]*Cos[4*t], -48*q^2*r*BesselJ[1, r]*Cos[4*t], -48*q^3*r*BesselJ[1,
r]*Cos[4*t], 6*q^3*r^3*BesselJ[1, r]*Cos[4*t]}
>
In[19]:=
Cases[el, _*Cos[6*_]]

Out[19]=
{-8*q^3*r^2*BesselJ[0, r]*Cos[6*t], -8*q^3*r*BesselJ[1, r]*Cos[6*t],
q^3*r^3*BesselJ[1, r]*Cos[6*t]}
>
Finally, here is the list of terms without Cos
>
In[21]:=
c0 = Complement[el, c1]

Out[21]=
{1536*BesselJ[0, r], -72*q^2*r^2*BesselJ[0, r], -80*q^3*r^2*BesselJ[0,
r], -384*q*r*BesselJ[1, r], -144*q^2*r*BesselJ[1, r],
-80*q^3*r*BesselJ[1, r], 10*q^3*r^3*BesselJ[1, r]}
>
> Hope this helps.
>
> Regards,
> Wolfgang

```

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