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A formula which expresses the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w0970401.png" /> as an [[Infinite product|infinite product]]:
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$#A+1 = 12 n = 0
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$#C+1 = 12 : ~/encyclopedia/old_files/data/W097/W.0907040 Wallis formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w0970402.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w0970403.png" /></td> </tr></table>
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A formula which expresses the number  $  \pi /2 $
 +
as an [[Infinite product|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.:
 
There exist other variants of this formula, e.g.:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w0970404.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \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 [[#References|[1]]] in his calculation of the area of a disc; it is one of the earliest examples of an infinite product.
 
Formula (1) was first employed by J. Wallis [[#References|[1]]] in his calculation of the area of a disc; it is one of the earliest examples of an infinite product.
Line 13: Line 56:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Wallis,  "Arithmetica infinitorum" , Oxford  (1656)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Wallis,  "Arithmetica infinitorum" , Oxford  (1656)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Formula (1) is a direct consequence of Euler's product formula
 
Formula (1) is a direct consequence of Euler's product formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w0970406.png" /></td> </tr></table>
+
$$
 +
\sin  z  = z \prod _ { n= } 1 ^  \infty  \left ( 1 -  
 +
\frac{z  ^ {2} }{n  ^ {2}
 +
\pi  ^ {2} }
 +
\right )
 +
$$
 +
 
 +
with  $  \pi /2 $.
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w0970407.png" />.
+
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
  
It can also be obtained by expressing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w0970408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w0970409.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w09704010.png" />, and by showing that
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w09704011.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w09704012.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097040/w09704013.png" />.
+
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====
 
====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>

Revision as of 08:28, 6 June 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 $ \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=13195
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article