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General Inverse Tangent Series of Unknown Names

Submitted by admin on Tue, 03/17/2009 - 4:58pm
General Inverse Tangent Series of Unknown Names
(Posted 08/02/2008)
 
The following general finite series formulas of unknown names have been established in connecting to the inverse tangent.
 
  • For any positive integer,
,
where and. If the value of y is outside of the range , add in the left side of (I), where m is integer.
 
 
Rewrite (I) in the expanded form
      .
 
 
  • Using the complex property tanh(ix) = tan(x) and tan(iy) = itanh(y) to the formula (I), gives
 
,
where, and .
 
 
Rewrite (II) in the expanded form
 
      .
 
 
  • Similarly, using tan(ix) = itanh(x) and substitute it into (II), gives
 
,
where , x > y and.
 
 
Rewrite (III) in the expanded form
      .
 
 
  • Using only tanh(ix) = itan(x) and substitute it into (I), gives
 
, 
where, x > y, , , and i is integer.
 
 
Rewrite (IV) in the expanded form
 
      .
 
 
 
  • The formulas (I), (II), (III), and (IV) can be used to derive innumerable series of variety forms. For instance, we obtain four Machin-like formulas when considering n = 1.
 
 
 
 
 
 
  • If we substitutein (V), we obtain
 
 
or
 
.
 
 
Rewrite (VI) in terms of tanh x/2 by using the identity .
 
.
 
 
  • By setting x = tanh x/2 in (VII), we obtain the Machin-like formula
 
.
 
 
Similarly, we get another formula for x < 0,
 
.
 
 
(Posted August 2, 2008)
 
 
In-Text or Website Citation
Tue N. Vu, General Inverse Tangent Series of Unknown Names, 08/02/2008, from Series Math Study Resource.
Hyperlink: http://www.seriesmathstudy.com/someatangents.htm.
 
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