RE: Integrate bug
- To: "mathgroup at yoda.ncsa.uiuc.edu"@gte.com
- Subject: RE: Integrate bug
- From: blachman%gtewd.dnet at gte.com (NELSON M. BLACHMAN)
- Date: Mon, 25 Feb 91 17:00:49 -0500
Mathematica (MS-DOS 386/7) 1.2 (September 27, 1989) [With pre-loaded data]
by S. Wolfram, D. Grayson, R. Maeder, H. Cejtin,
S. Omohundro, D. Ballman and J. Keiper
with I. Rivin, D. Withoff and T. Sherlock
Copyright 1988,1989 Wolfram Research Inc.
In[1]:= Integrate[E^(I x) Cos[x], {x, 0, 2Pi}]
Out[1]= 0 (Wrong!)
In[2]:= Integrate[Cos[ x] Cos[x], {x, 0, 2Pi}]
Out[2]= Pi (Right!)
In[3]:= Integrate[Sin[ x] Cos[x], {x, 0, 2Pi}]
Out[3]= 0 (Right!)
In[4]:= Integrate[(Cos[x] + I Sin[x]) Cos[x], {x, 0, 2Pi}]
Out[4]= Pi (Right!)
In[5]:= Integrate[E^(I x) (E^(I x) + E^(-I x))/2, {x, 0, 2 Pi}]
Out[5]= 0 (Wrong!)
In[13]:= Integrate[E^(I x) Cos[x], x]
-I 2 I x -I I x
Out[13]= -- E + -- Log[E ] (Right but absurd!)
4 2
Mathematica's error in evaluating the definite integral evidently stems from
evaluating this absurd logarithm on the wrong Riemann sheet. Evidently Mma
notices that E^{I x} d E^{I x}
E^{I x} E^{-I x} = ------- = - I --------- / E^{I x},
E^{I x} dx
thus finding the indefinite integral of this expression to be -I Log[E^{I x}]
without ever noticing the cancellation that was possible at the very start.
Nelson M. Blachman
GTE Government Systems Corp.
Mountain View, California
blachman%gtewd.dnet at gte.com