Re: Arctan rational
- From: Dave Seaman <dseaman@xxxxxxxxxxxx>
- Date: Mon, 31 Mar 2008 12:36:39 +0000 (UTC)
On Sun, 30 Mar 2008 23:56:22 -0500, quasi wrote:
Let's try Maple.
As a trivial check of prerequisites, I tried
> x:=arctan(1);
result: Pi/4
Note -- Maple simplifies it automatically.
Next, a warmup ...
> x:=arctan(1/2) + arctan(1/3);
> simplify(x);
result: Pi/4
Although Maple had to be asked to simplify it, the result is fine.
Next, the real problem ...
> x:=arctan(1/2) + arctan(1/5) + arctan(1/8);
> simplify(x);
result: arctan(1/2) + arctan(1/5) + arctan(1/8)
In other words, Maple has no clue on this one.
Had it succeeded, I would have let it have a shot at the expression
Robert Israel posted, the one which equals 5*Pi/4.
But since Maple couldn't handle the above sum with 3 terms, there's
essentially no chance for it to handle Robert's more complicated
expression. Since I knew it would fail, I didn't bother trying -- I
wouldn't want to hurt Maple's feelings.
How about Mathematica or Maxima?
Can either of those contenders simplify
arctan(1/2) + arctan(1/5) + arctan(1/8)
without being led by the hand?
I'll try Mathematica.
Mathematica 6.0 for Mac OS X x86 (64-bit)
Copyright 1988-2008 Wolfram Research, Inc.
In[1]:= ArcTan[1/2]+ArcTan[1/3]
1 1
Out[1]= ArcTan[-] + ArcTan[-]
3 2
In[2]:= Simplify[%]
1 1
Out[2]= ArcTan[-] + ArcTan[-]
3 2
In[3]:= FullSimplify[%]
Pi
Out[3]= --
4
In[4]:= FullSimplify[ArcTan[1/2]+ArcTan[1/5]+ArcTan[1/8]]
Pi
Out[4]= --
4
In[5]:= FullSimplify[ArcTan[1]+ ArcTan[1/2] + ArcTan[1/5] + ArcTan[1/8]
+ ArcTan[1/3]+ArcTan[1/7]+ArcTan[1/6]+ArcTan[1/31]+ArcTan[1/9]+ArcTan[1/73]
+ ArcTan[1/4]+ArcTan[1/10]+ArcTan[1/12]+ArcTan[1/13]+ArcTan[1/14]
+ArcTan[1/17]+ArcTan[1/21]+ArcTan[1/31]+ArcTan[1/43]+ArcTan[1/57]
+ArcTan[1/78]+ArcTan[1/91]+ArcTan[1/183]
+ ArcTan[1/11]+ArcTan[1/15]+ArcTan[1/18]+ArcTan[1/19]+ArcTan[1/22]
+ArcTan[1/23]+ArcTan[1/24]+ArcTan[1/25]+ArcTan[1/27]+ArcTan[1/28]
+ArcTan[1/30]+ArcTan[1/32]+ArcTan[1/41]+ArcTan[1/44]+ArcTan[1/46]
+ArcTan[1/47]+ArcTan[1/58]+ArcTan[1/74]+ArcTan[1/75]+ArcTan[1/83]
+ArcTan[1/92]+ArcTan[1/111]+ArcTan[1/119]+ArcTan[1/157]+ArcTan[1/162]
+ArcTan[1/184]+ArcTan[1/211]+ArcTan[1/242]+ArcTan[1/288]+ArcTan[1/463]
+ArcTan[1/553]+ArcTan[1/757]+ArcTan[1/993]+ArcTan[1/1139]+ArcTan[1/1893]
+ArcTan[1/3307]+ArcTan[1/5403]+ArcTan[1/8373]+ArcTan[1/33673]]
5 Pi
Out[5]= ----
4
In[6]:=
--
Dave Seaman
Court affirms Judge Yohn's ruling.
<http://www.ipsnews.net/news.asp?idnews=41761>
.
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- Arctan rational
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