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Difference between revisions of "Wallis formula"

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m (tex encoded by computer)
m (fix tex)
 
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\frac{4}{5}
 
\frac{4}{5}
 
  } \right ) \dots \left ( {
 
  } \right ) \dots \left ( {
\frac{2k}{2k-}
+
\frac{2k}{2k-1}
  1 } \cdot
+
  } \cdot
 
{
 
{
\frac{2k}{2k+}
+
\frac{2k}{2k+1}
  1 } \right ) \dots =
+
  } \right ) \dots =
 
$$
 
$$
  
 
$$  
 
$$  
 
= \  
 
= \  
\lim\limits _ {m \rightarrow \infty }  \prod _ { k= } 1 ^ { m }   
+
\lim\limits _ {m \rightarrow \infty }  \prod _ { k=1 } ^ { m }   
 
\frac{( 2k)  ^ {2} }{( 2k- 1)( 2k+ 1) }
 
\frac{( 2k)  ^ {2} }{( 2k- 1)( 2k+ 1) }
 
  .
 
  .
Line 61: Line 61:
  
 
$$  
 
$$  
\sin  z  =  z \prod _ { n= } 1 ^  \infty  \left ( 1 -  
+
\sin  z  =  z \prod _ { n=1 } ^  \infty  \left ( 1 - \frac{z  ^ {2} }{n  ^ {2} \pi  ^ {2} } \right )
\frac{z  ^ {2} }{n  ^ {2}
 
\pi  ^ {2} }
 
\right )
 
 
$$
 
$$
 
+
with  $z = \pi /2 $.
with  $ \pi /2 $.
 
  
 
It can also be obtained by expressing  $  \int _ {0} ^ {\pi /2 } \sin  ^ {2m}  x  dx $
 
It can also be obtained by expressing  $  \int _ {0} ^ {\pi /2 } \sin  ^ {2m}  x  dx $
and  $  \int _ {0} ^ {\pi /2 } \sin  ^ {2m+} 1 x  dx $
+
and  $  \int _ {0} ^ {\pi /2 } \sin  ^ {2m+1}  x  dx $ in terms of  $  m $, and by showing that
in terms of  $  m $,  
 
and by showing that
 
  
 
$$  
 
$$  
  
\frac{\int\limits _ { 0 } ^ {  \pi  /2 } \sin  ^ {2m}  x  dx }{\int\limits _ { 0 } ^ {  \pi  /2 } \sin  ^ {2m+} 1 x  dx }
+
\frac{\int\limits _ { 0 } ^ {  \pi  /2 } \sin  ^ {2m}  x  dx }{\int\limits _ { 0 } ^ {  \pi  /2 } \sin  ^ {2m+1}  x  dx }
 
   \rightarrow  1 \  ( m\rightarrow \infty ).
 
   \rightarrow  1 \  ( m\rightarrow \infty ).
 
$$
 
$$
  
Formula (2) can be derived from (1) by multiplying the numerator and the denominator of  $  \prod _ {k=} 1 ^ {m} ( 2k)  ^ {2} / ( 2k- 1)( 2k+ 1) $
+
Formula (2) can be derived from (1) by multiplying the numerator and the denominator of  $  \prod _ {k=1}  ^ {m} ( 2k)  ^ {2} / ( 2k- 1)( 2k+ 1) $
 
by  $  2  ^ {2} \cdot 4  ^ {2} \dots ( 2m)  ^ {2} $.
 
by  $  2  ^ {2} \cdot 4  ^ {2} \dots ( 2m)  ^ {2} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Calculus" , '''2''' , Blaisdell  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.H. Edwards jr.,  "The historical development of the calculus" , Springer  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Lax,  S. Burstein,  A. Lax,  "Calculus with applications and computing" , '''1''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.J. Struik (ed.) , ''A source book in mathematics: 1200–1800'' , Harvard Univ. Press  (1986)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Calculus" , '''2''' , Blaisdell  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.H. Edwards jr.,  "The historical development of the calculus" , Springer  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Lax,  S. Burstein,  A. Lax,  "Calculus with applications and computing" , '''1''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.J. Struik (ed.) , ''A source book in mathematics: 1200–1800'' , Harvard Univ. Press  (1986)</TD></TR></table>

Latest revision as of 21:31, 29 December 2020


A formula which expresses the number $ \pi /2 $ as an infinite product:

$$ \tag{1 } { \frac \pi {2} } = \ \left ( { \frac{2}{1} } \cdot { \frac{2}{3} } \right ) \left ( { \frac{4}{3} } \cdot { \frac{4}{5} } \right ) \dots \left ( { \frac{2k}{2k-1} } \cdot { \frac{2k}{2k+1} } \right ) \dots = $$

$$ = \ \lim\limits _ {m \rightarrow \infty } \prod _ { k=1 } ^ { m } \frac{( 2k) ^ {2} }{( 2k- 1)( 2k+ 1) } . $$

There exist other variants of this formula, e.g.:

$$ \tag{2 } \sqrt \pi = \ \lim\limits _ {m \rightarrow \infty } \ \frac{( m!) ^ {2} \cdot 2 ^ {2m} }{( 2m)! \sqrt m } . $$

Formula (1) was first employed by J. Wallis [1] in his calculation of the area of a disc; it is one of the earliest examples of an infinite product.

References

[1] J. Wallis, "Arithmetica infinitorum" , Oxford (1656)

Comments

Formula (1) is a direct consequence of Euler's product formula

$$ \sin z = z \prod _ { n=1 } ^ \infty \left ( 1 - \frac{z ^ {2} }{n ^ {2} \pi ^ {2} } \right ) $$ with $z = \pi /2 $.

It can also be obtained by expressing $ \int _ {0} ^ {\pi /2 } \sin ^ {2m} x dx $ and $ \int _ {0} ^ {\pi /2 } \sin ^ {2m+1} x dx $ in terms of $ m $, and by showing that

$$ \frac{\int\limits _ { 0 } ^ { \pi /2 } \sin ^ {2m} x dx }{\int\limits _ { 0 } ^ { \pi /2 } \sin ^ {2m+1} x dx } \rightarrow 1 \ ( m\rightarrow \infty ). $$

Formula (2) can be derived from (1) by multiplying the numerator and the denominator of $ \prod _ {k=1} ^ {m} ( 2k) ^ {2} / ( 2k- 1)( 2k+ 1) $ by $ 2 ^ {2} \cdot 4 ^ {2} \dots ( 2m) ^ {2} $.

References

[a1] T.M. Apostol, "Calculus" , 2 , Blaisdell (1964)
[a2] C.H. Edwards jr., "The historical development of the calculus" , Springer (1979)
[a3] P. Lax, S. Burstein, A. Lax, "Calculus with applications and computing" , 1 , Springer (1976)
[a4] D.J. Struik (ed.) , A source book in mathematics: 1200–1800 , Harvard Univ. Press (1986)
How to Cite This Entry:
Wallis formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wallis_formula&oldid=51104
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article