Re: Mathematica bug in handling trigonometric functions? (and more)

• To: mathgroup at smc.vnet.net
• Subject: [mg55755] Re: Mathematica bug in handling trigonometric functions? (and more)
• From: dh <dh at metrohm.ch>
• Date: Tue, 5 Apr 2005 05:45:06 -0400 (EDT)
• References: <d2oes5\$fhn\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Hi APC,
I tried to simplify the problem a bit. There is definitly a bug that
Wolfram should take notice. It would be nice if WRI could give an answer .

The folllowing is obviously correct:
Sum[Sin[k]*Cos[k + 1], {k, 1, 1}]
Out: Cos[2] Sin[1]

When we do the same with indefinite summation:
Sum[Sin[k]*Cos[k + 1], {k, 1, n}] /. n -> 1 // Simplify // Expand
Out: Sin[1]/2 + Cos[2] Sin[1]

we get an additional term: Sin[1]/2 !!!

Sincerely, Daniel

APC wrote:
> a)"Sum"-Command
>
> mathematica 5.1:
>
> input:   Sum[Sin[k]*Cos[k + 1], {k, 1, n}]
> output:  (1/4)*(Cos[2] - Cos[2 + 2*n])*Csc[1]
>
> which is obviously incorrect.
>
> with the more general Sum[Sin[a*k]*Cos[k + 1], {k, 1, n}] i get correct results, though.
>
> (mathematica 2.2's "SymbolicSum" command has no problems at all)
>
> another example:
>
> input:  Sum[Log[k]/Exp[k], {k, 1, Infinity}]
> output:  (-(E*Derivative[1, 0][PolyLog][0, E^(-1)]) +
>   Derivative[0, 1, 0][LerchPhi][E^(-1), 0, 1])/E
>
> which is incorrect. -Derivative[1, 0][PolyLog][0, E^(-1)]alone is the correct result, the LerchPhi term is identical so the output could be simplified to "0", which is wrong, of course. there seems to be a problem with "Sum" and derivatives of functions, as
>
> Sum[Log[k]^2/k^3, {k, 1, Infinity}]
>
> for example yields an incorrect result under similar circumstances.
>
> b)"Integrate" command
>
> input:  Integrate[Sqrt[Sin[x] + Cos[x]], x]
> output:  (2*(Cos[x] + Sin[x] + Null*(1 + Sin[2*x])^(1/4)))/Sqrt[Cos[x] + Sin[x]]
>
> the output includes a "Null"-term.
>
> the result for the more general case
>
> input:  Integrate[Sqrt[a*Sin[x] + b*Cos[x]], x]
>
> is correct, though.
>
> apparently "Sum" and "Integrate" have problems handling special cases involving trigonometric functions.
> please check for yourself. if i'm correct, i can no more trust any "Sum" or "Integrate" results without checking twice.
>

```

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