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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054330/j0543301.png" /> be a regular [[Analytic function|analytic function]] of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054330/j0543302.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054330/j0543303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054330/j0543304.png" />, up to a discrete set of singular points. If there is a sequence of semi-circles
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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/j/j054/j054330/j0543305.png" /></td> </tr></table>
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such that the maximum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054330/j0543306.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054330/j0543307.png" /> tends to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054330/j0543308.png" />, then
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Let  $  f( z) $
 +
be a regular [[Analytic function|analytic function]] of a complex variable  $  z $,
 +
where  $  | z | > c \geq  0 $,
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$  \mathop{\rm Im}  z \geq  0 $,
 +
up to a discrete set of singular points. If there is a sequence of semi-circles
  
<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/j/j054/j054330/j0543309.png" /></td> </tr></table>
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$$
 +
\gamma ( R _ {n} )  = \{ {z } : {| z | = R _ {n} ,\
 +
\mathop{\rm Im}  z \geq  0 } \}
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,\  R _ {n} \uparrow + \infty ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054330/j05433010.png" /> is any positive number. Jordan's lemma can be applied to residues not only under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054330/j05433011.png" />, but even when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054330/j05433012.png" /> uniformly on a sequence of semi-circles in the upper or lower half-plane. For example, in order to calculate integrals of the form
+
such that the maximum  $  M ( R _ {n} ) = \max  | f ( z) | $
 +
on $  \gamma ( R _ {n} ) $
 +
tends to zero as  $  n \rightarrow \infty $,  
 +
then
  
<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/j/j054/j054330/j05433013.png" /></td> </tr></table>
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$$
 +
\lim\limits _ {n \rightarrow \infty }  \int\limits _ {\gamma ( R _ {n} ) }
 +
e  ^ {iaz} f ( z)  dz  = 0 ,
 +
$$
 +
 
 +
where  $  a $
 +
is any positive number. Jordan's lemma can be applied to residues not only under the condition  $  zf ( z) \rightarrow 0 $,
 +
but even when  $  f ( z) \rightarrow 0 $
 +
uniformly on a sequence of semi-circles in the upper or lower half-plane. For example, in order to calculate integrals of the form
 +
 
 +
$$
 +
\int\limits _ {- \infty } ^  \infty  f ( x)  \cos  a x \
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d x ,\  \int\limits _ {- \infty } ^  \infty  f ( x)  \sin  a x  d x .
 +
$$
  
 
Obtained by C. Jordan [[#References|[1]]].
 
Obtained by C. Jordan [[#References|[1]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Jordan,  "Cours d'analyse" , '''2''' , Gauthier-Villars  (1894)  pp. 285–286</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 6</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Jordan,  "Cours d'analyse" , '''2''' , Gauthier-Villars  (1894)  pp. 285–286</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 6</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.S. Mitrinović,  J.D. Kečkić,  "The Cauchy method of residues: theory and applications" , Reidel  (1984)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.S. Mitrinović,  J.D. Kečkić,  "The Cauchy method of residues: theory and applications" , Reidel  (1984)</TD></TR></table>

Latest revision as of 22:14, 5 June 2020


Let $ f( z) $ be a regular analytic function of a complex variable $ z $, where $ | z | > c \geq 0 $, $ \mathop{\rm Im} z \geq 0 $, up to a discrete set of singular points. If there is a sequence of semi-circles

$$ \gamma ( R _ {n} ) = \{ {z } : {| z | = R _ {n} ,\ \mathop{\rm Im} z \geq 0 } \} ,\ R _ {n} \uparrow + \infty , $$

such that the maximum $ M ( R _ {n} ) = \max | f ( z) | $ on $ \gamma ( R _ {n} ) $ tends to zero as $ n \rightarrow \infty $, then

$$ \lim\limits _ {n \rightarrow \infty } \int\limits _ {\gamma ( R _ {n} ) } e ^ {iaz} f ( z) dz = 0 , $$

where $ a $ is any positive number. Jordan's lemma can be applied to residues not only under the condition $ zf ( z) \rightarrow 0 $, but even when $ f ( z) \rightarrow 0 $ uniformly on a sequence of semi-circles in the upper or lower half-plane. For example, in order to calculate integrals of the form

$$ \int\limits _ {- \infty } ^ \infty f ( x) \cos a x \ d x ,\ \int\limits _ {- \infty } ^ \infty f ( x) \sin a x d x . $$

Obtained by C. Jordan [1].

References

[1] C. Jordan, "Cours d'analyse" , 2 , Gauthier-Villars (1894) pp. 285–286
[2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1967) (In Russian)
[3] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6

Comments

References

[a1] D.S. Mitrinović, J.D. Kečkić, "The Cauchy method of residues: theory and applications" , Reidel (1984)
How to Cite This Entry:
Jordan lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_lemma&oldid=47469
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article