Difference between revisions of "Bürmann-Lagrange series"
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''Lagrange series'' | ''Lagrange series'' | ||
− | A power series which offers a complete solution to the problem of local inversion of holomorphic functions. In fact, let a function | + | A power series which offers a complete solution to the problem of local inversion of holomorphic functions. In fact, let a function $ w = g(z) $ |
+ | of the complex variable $ z $ | ||
+ | be regular in a neighbourhood of the point $ z = a $, | ||
+ | and let $ g(a) = b $ | ||
+ | and $ g ^ \prime (a) \neq 0 $. | ||
+ | Then there exists a regular function $ z = h (w) $ | ||
+ | in some neighbourhood of the point $ w = b $ | ||
+ | of the $ w $- | ||
+ | plane which is the inverse to $ g(z) $ | ||
+ | and is such that $ h(b) = a $. | ||
+ | Moreover, if $ f(z) $ | ||
+ | is any regular function in a neighbourhood of the point $ z = a $, | ||
+ | then the composite function $ F(w) = f[h(w)] $ | ||
+ | can be expanded in a Bürmann–Lagrange series in a neighbourhood of the point $ w = b $ | ||
− | + | $$ \tag{* } | |
+ | F (w) = \ | ||
+ | f (a) + | ||
+ | $$ | ||
− | + | $$ | |
+ | + | ||
+ | \sum _ {n = 1 } ^ \infty { | ||
+ | \frac{1}{n!} | ||
+ | } \left \{ | ||
+ | \frac{d ^ {n - 1 } }{dz ^ {n - 1 } } | ||
− | + | \left [ f ^ { \prime } (z) \left ( | |
+ | \frac{z - a }{g (z) - b } | ||
− | + | \right ) ^ {n} \right ] \right \} _ {z = a } (w - b) ^ {n} . | |
+ | $$ | ||
− | + | The inverse of the function $ w = g(z) $ | |
+ | is obtained by setting $ f(z) \equiv z $. | ||
− | + | The expansion (*) follows from Bürmann's theorem [[#References|[1]]]: Under the assumptions made above on the holomorphic functions $ g(z) $ | |
+ | and $ f(z) $, | ||
+ | the latter function may be represented in a certain domain in the $ z $- | ||
+ | plane containing $ a $ | ||
+ | in the form | ||
+ | |||
+ | $$ | ||
+ | f (z) = f (a) + | ||
+ | $$ | ||
+ | |||
+ | $$ | ||
+ | + | ||
+ | \sum _ {n = 1 } ^ { {m } - 1 } | ||
+ | \frac{[g (z) - b] ^ {n} }{n!} | ||
+ | \left \{ | ||
+ | \frac{d ^ {n - 1 } }{dz ^ {n - 1 } } | ||
+ | \left [ | ||
+ | f ^ { \prime } (z) \left ( | ||
+ | \frac{z - a }{g (z) - b } | ||
+ | \right ) ^ {n} \right ] \right \} _ {z = a } + R _ {m} , | ||
+ | $$ | ||
where | where | ||
− | + | $$ | |
+ | R _ {m} = | ||
+ | \frac{1}{2 \pi i } | ||
+ | |||
+ | \int\limits _ { a } ^ { z } \int\limits _ \gamma | ||
+ | \left [ | ||
+ | |||
+ | \frac{g (z) - b }{g (t) - b } | ||
− | + | \right ] ^ {m - 1 } | |
− | + | \frac{f ^ { \prime } (t) g ^ \prime (z) dt dz }{g (t) - g (z) } | |
+ | . | ||
+ | $$ | ||
− | + | Here $ \gamma $ | |
+ | is a contour in the $ t $- | ||
+ | plane which encloses the points $ a $ | ||
+ | and $ z $, | ||
+ | and is such that if $ \zeta $ | ||
+ | is any point inside $ \gamma $, | ||
+ | then the equation $ g(t) = g( \zeta ) $ | ||
+ | has no roots on $ \gamma $ | ||
+ | or inside $ \gamma $ | ||
+ | other than the simple root $ t = \zeta $. | ||
− | + | The expansion (*) for the case $ b = 0 $ | |
+ | was obtained by J.L. Lagrange . | ||
− | + | If the derivative $ g ^ \prime (t) $ | |
+ | has a zero of order $ r - 1 $ | ||
+ | at the point $ z = a $, | ||
+ | there is the following generalization of the Bürmann–Lagrange series for the multi-valued inverse function [[#References|[3]]]: | ||
− | Another generalization (see, for example, [[#References|[4]]]) refers to functions | + | $$ |
+ | F (w) = f (a) + | ||
+ | $$ | ||
+ | |||
+ | $$ | ||
+ | + | ||
+ | \sum _ {n = 1 } ^ \infty { | ||
+ | \frac{1}{n!} | ||
+ | } \left \{ | ||
+ | \frac{d ^ {n - 1 } }{dz ^ {n - 1 } } | ||
+ | \left [ f ^ | ||
+ | { \prime } (z) \left ( | ||
+ | \frac{z - a }{g (z) - b } | ||
+ | \right ) | ||
+ | ^ {n} \right ] \right \} _ {z = a } (w - b) ^ {n/r} . | ||
+ | $$ | ||
+ | |||
+ | Another generalization (see, for example, [[#References|[4]]]) refers to functions $ g(z) $ | ||
+ | regular in an annulus; instead of the series (*), one obtains a series with positive and negative powers of the difference $ w - b $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Bürmann, ''Mem. Inst. Nat. Sci. Arts. Sci. Math. Phys.'' , '''2''' (1799) pp. 13–17</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> J.L. Lagrange, ''Mem. Acad. R. Sci. et Belles-lettres Berlin'' , '''24''' (1770)</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> J.L. Lagrange, "Additions au mémoire sur la résolution des équations numériques" , ''Oeuvres'' , '''2''' , G. Olms (1973) pp. 579–652</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer (1968) pp. Chapt. 7</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Bürmann, ''Mem. Inst. Nat. Sci. Arts. Sci. Math. Phys.'' , '''2''' (1799) pp. 13–17</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> J.L. Lagrange, ''Mem. Acad. R. Sci. et Belles-lettres Berlin'' , '''24''' (1770)</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> J.L. Lagrange, "Additions au mémoire sur la résolution des équations numériques" , ''Oeuvres'' , '''2''' , G. Olms (1973) pp. 579–652</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer (1968) pp. Chapt. 7</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== |
Latest revision as of 06:29, 30 May 2020
Lagrange series
A power series which offers a complete solution to the problem of local inversion of holomorphic functions. In fact, let a function $ w = g(z) $ of the complex variable $ z $ be regular in a neighbourhood of the point $ z = a $, and let $ g(a) = b $ and $ g ^ \prime (a) \neq 0 $. Then there exists a regular function $ z = h (w) $ in some neighbourhood of the point $ w = b $ of the $ w $- plane which is the inverse to $ g(z) $ and is such that $ h(b) = a $. Moreover, if $ f(z) $ is any regular function in a neighbourhood of the point $ z = a $, then the composite function $ F(w) = f[h(w)] $ can be expanded in a Bürmann–Lagrange series in a neighbourhood of the point $ w = b $
$$ \tag{* } F (w) = \ f (a) + $$
$$ + \sum _ {n = 1 } ^ \infty { \frac{1}{n!} } \left \{ \frac{d ^ {n - 1 } }{dz ^ {n - 1 } } \left [ f ^ { \prime } (z) \left ( \frac{z - a }{g (z) - b } \right ) ^ {n} \right ] \right \} _ {z = a } (w - b) ^ {n} . $$
The inverse of the function $ w = g(z) $ is obtained by setting $ f(z) \equiv z $.
The expansion (*) follows from Bürmann's theorem [1]: Under the assumptions made above on the holomorphic functions $ g(z) $ and $ f(z) $, the latter function may be represented in a certain domain in the $ z $- plane containing $ a $ in the form
$$ f (z) = f (a) + $$
$$ + \sum _ {n = 1 } ^ { {m } - 1 } \frac{[g (z) - b] ^ {n} }{n!} \left \{ \frac{d ^ {n - 1 } }{dz ^ {n - 1 } } \left [ f ^ { \prime } (z) \left ( \frac{z - a }{g (z) - b } \right ) ^ {n} \right ] \right \} _ {z = a } + R _ {m} , $$
where
$$ R _ {m} = \frac{1}{2 \pi i } \int\limits _ { a } ^ { z } \int\limits _ \gamma \left [ \frac{g (z) - b }{g (t) - b } \right ] ^ {m - 1 } \frac{f ^ { \prime } (t) g ^ \prime (z) dt dz }{g (t) - g (z) } . $$
Here $ \gamma $ is a contour in the $ t $- plane which encloses the points $ a $ and $ z $, and is such that if $ \zeta $ is any point inside $ \gamma $, then the equation $ g(t) = g( \zeta ) $ has no roots on $ \gamma $ or inside $ \gamma $ other than the simple root $ t = \zeta $.
The expansion (*) for the case $ b = 0 $ was obtained by J.L. Lagrange .
If the derivative $ g ^ \prime (t) $ has a zero of order $ r - 1 $ at the point $ z = a $, there is the following generalization of the Bürmann–Lagrange series for the multi-valued inverse function [3]:
$$ F (w) = f (a) + $$
$$ + \sum _ {n = 1 } ^ \infty { \frac{1}{n!} } \left \{ \frac{d ^ {n - 1 } }{dz ^ {n - 1 } } \left [ f ^ { \prime } (z) \left ( \frac{z - a }{g (z) - b } \right ) ^ {n} \right ] \right \} _ {z = a } (w - b) ^ {n/r} . $$
Another generalization (see, for example, [4]) refers to functions $ g(z) $ regular in an annulus; instead of the series (*), one obtains a series with positive and negative powers of the difference $ w - b $.
References
[1] | H. Bürmann, Mem. Inst. Nat. Sci. Arts. Sci. Math. Phys. , 2 (1799) pp. 13–17 |
[2a] | J.L. Lagrange, Mem. Acad. R. Sci. et Belles-lettres Berlin , 24 (1770) |
[2b] | J.L. Lagrange, "Additions au mémoire sur la résolution des équations numériques" , Oeuvres , 2 , G. Olms (1973) pp. 579–652 |
[3] | A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1968) pp. Chapt. 7 |
[4] | E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6 |
[5] | A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) |
Comments
There is an exhaustive treatment of the Lagrange–Bürmann theorem and series in [a1].
References
[a1] | P. Henrici, "Applied and computational complex analysis" , 1 , Wiley (1974) |
Bürmann-Lagrange series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=B%C3%BCrmann-Lagrange_series&oldid=23202