Mathematica - SeriesCoefficient: needs work!

Author

SeriesCoefficient: needs work!

jackgold@umich.edu

2007-11-19, 8:18 am

Hi Everyone,

I have been experimenting with the new (ver 6) command,
SeriesCoefficient in the form,

SeriesCoefficient[fnt,{x,x0,n}].

Here fnt is a function of x and n is symbolic. This command is
supposed to return the nth coefficient in the series expansion of fnt
about x0.

I have found the following results on a MacBook Pro running Tiger.

1) SeriesCoefficient[Cos[x] Exp[x], {x, 0, n}] returns itself, unevaluated.

2) SeriesCoefficient[Cos[x] Exp[x]/(1-x), {x, 0, n}] returns an
expression involving incomplete Gamma functions with an imaginary
argument. Odd that 1) does not compute but 2) does! Not that the
result is terribly revealing, by the way.

3) SeriesCoefficient[Sin[x] Exp[x], {x, 0, n}] so preposterously
complicated that most of us would have preferred no result! (Just
joking. The result is far to complicated to publish here.)

My opinion is that this use of SeriesCoefficient should not be offered
to the public until some of these obvious glitches are cleaned up.
Incidentally, since the nth terms of the individual functions Sin[x],
Cos[x], Exp[x] and 1/(1-x) can be found using SeriesCoefficient and
surely Mathematica knows how to find the nth coefficient of a product of power
series, I suspect the problem lies in the finite summation which
results from the use of the Cauchy product formula.

Peter Pein

2007-11-21, 4:42 am

jackgold@umich.edu schrieb:
> Hi Everyone,
>
> I have been experimenting with the new (ver 6) command,
> SeriesCoefficient in the form,
>
> SeriesCoefficient[fnt,{x,x0,n}].
>
> Here fnt is a function of x and n is symbolic. This command is
> supposed to return the nth coefficient in the series expansion of fnt
> about x0.
>
> I have found the following results on a MacBook Pro running Tiger.
>
> 1) SeriesCoefficient[Cos[x] Exp[x], {x, 0, n}] returns itself, unevaluated.
>
> 2) SeriesCoefficient[Cos[x] Exp[x]/(1-x), {x, 0, n}] returns an
> expression involving incomplete Gamma functions with an imaginary
> argument. Odd that 1) does not compute but 2) does! Not that the
> result is terribly revealing, by the way.
>
> 3) SeriesCoefficient[Sin[x] Exp[x], {x, 0, n}] so preposterously
> complicated that most of us would have preferred no result! (Just
> joking. The result is far to complicated to publish here.)
>
> My opinion is that this use of SeriesCoefficient should not be offered
> to the public until some of these obvious glitches are cleaned up.
> Incidentally, since the nth terms of the individual functions Sin[x],
> Cos[x], Exp[x] and 1/(1-x) can be found using SeriesCoefficient and
> surely Mathematica knows how to find the nth coefficient of a product of power
> series, I suspect the problem lies in the finite summation which
> results from the use of the Cauchy product formula.
>
>
Strange.
In Version 5.2 the good old SeriesTerm[] in the package DiscreteMath got its
difficulties in case 2:

In[1]:=
<< "DiscreteMath`"
In[2]:=
SeriesTerm[Cos[x]*E^x, {x, 0, n}]
Out[2]=
(1 - I)^n/(2*n!) + (1 + I)^n/(2*n!)
In[3]:=
SeriesTerm[(Cos[x]*E^x)/(1 - x),
{x, 0, n}]
Out[3]=
Piecewise[{{ComplexInfinity, n >= 1}},
(I^n*KroneckerDelta[Mod[n, 2]])/
Gamma[1 + n]]
In[4]:=
SeriesTerm[Sin[x]*E^x, {x, 0, n}]
Out[4]=
-((1/Pi)*(2^(1 + n/2)*
(Cos[(n*Pi)/4] + Cos[(3*n*Pi)/4])*
Gamma[-n]*Sin[(n*Pi)/4]^2))

And the package SpecialFunctions by Prof. Wolfram Koepf (
http://www.mathematik.uni-kassel.de...f/CA/index.html ) has got problems
with case 2 too:

In[1]:=
<< "SpecialFunctions`"
>From In[1]:=
"SpecialFunctions, (C) Wolfram Koepf, version 2.01, 2006"
>From In[1]:=
"Fast Zeilberger, (C) Peter Paule and Markus Schorn (V 2.2) loaded"
In[3]:=
PS[Cos[x]*Exp[x], x, 0]
Out[3]=
sum[(2^(k/2)*x^k*Cos[(k*Pi)/4])/k!, {k, 0, Infinity}]
In[4]:=
PS[Cos[x]*(Exp[x]/(1 - x)), x, 0]
Out[4]=
SpecialFunctions`Private`df[3]
In[5]:=
PS[Sin[x]*Exp[x], x, 0]
Out[5]=
sum[(2^(k/2)*x^k*Sin[(k*Pi)/4])/k!, {k, 0, Infinity}]

Shouldn't the V6-function SeriesCoefficient be at least as smart as the
V5.2-package is? But maybe the new version gets some series representations
where the old one fails.

Peter

Devendra Kapadia

2007-11-22, 4:40 am

On Wed, 21 Nov 2007, Peter Pein wrote:

> jackgold@umich.edu schrieb:
> Strange.
> In Version 5.2 the good old SeriesTerm[] in the package DiscreteMath got its
> difficulties in case 2:
>
> In[1]:=
> << "DiscreteMath`"
> In[2]:=
> SeriesTerm[Cos[x]*E^x, {x, 0, n}]
> Out[2]=
> (1 - I)^n/(2*n!) + (1 + I)^n/(2*n!)
> In[3]:=
> SeriesTerm[(Cos[x]*E^x)/(1 - x),
> {x, 0, n}]
> Out[3]=
> Piecewise[{{ComplexInfinity, n >= 1}},
> (I^n*KroneckerDelta[Mod[n, 2]])/
> Gamma[1 + n]]
> In[4]:=
> SeriesTerm[Sin[x]*E^x, {x, 0, n}]
> Out[4]=
> -((1/Pi)*(2^(1 + n/2)*
> (Cos[(n*Pi)/4] + Cos[(3*n*Pi)/4])*
> Gamma[-n]*Sin[(n*Pi)/4]^2))
>
> And the package SpecialFunctions by Prof. Wolfram Koepf (
> http://www.mathematik.uni-kassel.de...f/CA/index.html ) has got problems
> with case 2 too:
>
> In[1]:=
> << "SpecialFunctions`"
> "SpecialFunctions, (C) Wolfram Koepf, version 2.01, 2006"
> "Fast Zeilberger, (C) Peter Paule and Markus Schorn (V 2.2) loaded"
> In[3]:=
> PS[Cos[x]*Exp[x], x, 0]
> Out[3]=
> sum[(2^(k/2)*x^k*Cos[(k*Pi)/4])/k!, {k, 0, Infinity}]
> In[4]:=
> PS[Cos[x]*(Exp[x]/(1 - x)), x, 0]
> Out[4]=
> SpecialFunctions`Private`df[3]
> In[5]:=
> PS[Sin[x]*Exp[x], x, 0]
> Out[5]=
> sum[(2^(k/2)*x^k*Sin[(k*Pi)/4])/k!, {k, 0, Infinity}]
>
>
> Shouldn't the V6-function SeriesCoefficient be at least as smart as the
> V5.2-package is? But maybe the new version gets some series representations
> where the old one fails.
>
> Peter
>
Hello,

Thank you for these examples and observations about SeriesCoefficient
in Mathematica 6.

The problem with the first and third examples (general series coefficients
for Cos[x]*Exp[x] and Sin[x]*Exp[x]) is indeed caused by difficulties in
evaluating the underlying finite sums. A partial workaround is to use
TrigToExp in these cases as shown below (for completeness, all three
examples are evaluated along with checks using the Series function).

==============================

In[1]:= $Version

Out[1]= 6.0 for Linux x86 (32-bit) (June 28, 2007)

In[2]:= SeriesCoefficient[Cos[x] Exp[x] // TrigToExp, {x, 0, n}]//InputForm

Out[2]//InputForm= (1 - I)^n/(2*n!) + (1 + I)^n/(2*n!)

In[3]:= Table[%, {n, 0, 7}] // InputForm

Out[3]//InputForm= {1, 1, 0, -1/3, -1/6, -1/30, 0, 1/630}

In[4]:= Normal[Series[Cos[x] Exp[x], {x, 0, 7}]] // InputForm

Out[4]//InputForm= 1 + x - x^3/3 - x^4/6 - x^5/30 + x^7/630

In[5]:= SeriesCoefficient[Cos[x] Exp[x]/(1 - x), {x, 0, n}] // InputForm

Out[5]//InputForm=
(E^(1 - I)*(1 + n)*Gamma[1 + n, 1 - I])/(2*Gamma[2 + n]) +
(E^(1 + I)*(1 + n)*Gamma[1 + n, 1 + I])/(2*Gamma[2 + n])

In[6]:= Table[%, {n, 0, 7}] // FullSimplify // InputForm

Out[6]//InputForm= {1, 2, 2, 5/3, 3/2, 22/15, 22/15, 185/126}

In[7]:= Normal[Series[Cos[x] Exp[x]/(1 - x), {x, 0, 7}]] // InputForm

Out[7]//InputForm=
1 + 2*x + 2*x^2 + (5*x^3)/3 + (3*x^4)/2 + (22*x^5)/15 + (22*x^6)/15 +
(185*x^7)/126

In[8]:= SeriesCoefficient[Sin[x] Exp[x] // TrigToExp, {x, 0, n}] // InputForm

Out[8]//InputForm= ((I/2)*(1 - I)^n)/n! - ((I/2)*(1 + I)^n)/n!

In[9]:= Table[%, {n, 0, 7}] // InputForm

Out[9]//InputForm= {0, 1, 1, 1/3, 0, -1/30, -1/90, -1/630}

In[10]:= Normal[Series[Sin[x] Exp[x] // TrigToExp, {x, 0, 7}]] //InputForm

Out[10]//InputForm= x + x^2 + x^3/3 - x^5/30 - x^6/90 - x^7/630

===============================

I apologize for the confusion caused by these problems.

Sincerely,

Devendra Kapadia,
Wolfram Research, Inc.

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