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Mathematica bug in handling trigonometric functions? (and more)
*To*: mathgroup at smc.vnet.net
*Subject*: [mg55683] Mathematica bug in handling trigonometric functions? (and more)
*From*: APC <caraman48 at yahoo.com>
*Date*: Sun, 3 Apr 2005 05:50:51 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
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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