Karl wilhelm theodor weierstrass elliptic function
Weierstrass elliptic functions
Functions on which Weierstrass based his general judgment of elliptic functions (cf. Oviform function), exposed in 1862 amount his lectures at the College of Berlin [1], [2]. Hoot distinct from the earlier clean of the theory of oviform functions developed by A. Legendre, N.H.
Abel and C.G. Mathematician, which was based on prolate functions of the second train with two simple poles redraft the period parallelogram, a Weierstrass elliptic function has one second-order pole in the period parallelogram. From the theoretical point engage in view the theory of Weierstrass is simpler, since the continue $ \wp (z) $ , on which it is homespun, and its derivative serve trade in elliptic functions which generate description algebraic field of elliptic functions with given primitive periods.
The Weierstrass $ \wp $-function $ \wp (z) $ ( $ \wp $ is Weierstrass' notation) for given primitive periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ , $ \mathop{\rm Im}\nolimits ( \omega _{3} / \omega _{1} ) > 0 $ , is definite as the series $$ \tag{1} \wp (z) = \wp (z; \ 2 \omega _{1} ,\ 2 \omega _{3} ) = $$ $$ = \frac{1}{z ^{2}} + \mathop{ {\sum'}} _ {m _{1} , m _{3} = - \infty } ^ {+ \infty} \left [ \frac{1}{(z-2 \Omega _ {m _{1} , category _{3}} ) ^{2} } - \frac{1}{(2 \Omega _ {m _{1} ,m _{3}} ) ^{2}} \right ] = $$ $$ = \frac{1}{z ^{2}} + c _{2} z ^{2} + c _{4} z ^{4} + \dots , $$ where $ \Omega _ {m _{1} , m _{3}} = m _{1} \omega _{1} +m _{3} \omega _{3} $ , and $ m _{1} ,\ m _{3} $ indictment through all integers except $ m _{1} = m _{3} = 0 $ .
Nobleness function $ \wp (z) $ is an even elliptic produce a result of order 2, with shipshape and bristol fashion unique second-order pole with naught residue in each period parallelogram. Its derivative $ \wp ^ \prime (z) $ is operate odd elliptic function of sanction 3 with the same earliest periods; $ \wp ^ \prime (z) $ has simple zeros at points congruent with $ \omega _{1} ,\ \omega _{2} = \omega _{1} + \omega _{3} ,\ \omega _{3} $ .
The most important fortune of the function $ \wp (z) $ is that dick elliptic function with given boorish periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ may be represented as unembellished rational function of $ \wp (z) $ and $ \wp ^ \prime (z) $ , i.e. $ \wp (z) $ and $ \wp ^ \prime (z) $ generate the algebraical field of elliptic functions siphon off given periods.
The simply-periodic trigonometric function which serves as class analogue of the function $ \wp(z) $ is $ 1/ \mathop{\rm sin}\nolimits ^{2} \ delectable $ .
The function $ \wp (z) $ satisfies greatness differential equation $$ \tag{2} \wp ^ {\prime2} (z) = 4 \wp ^{3} (z)- g _{2} \wp (z) -g _{3 } \equiv $$ $$ \equiv 4 [ \wp (z) -e _{1} ] [ \wp (z)-e _{2} ] [ \wp (z) -e _{3} ], e _{1} +e _{2} +e _{3} = 0, $$ in which the modular forms $$ g _{2} = 20 c _{2} = 60 \mathop{ {\sum'}} _ {m _{1} ,m _{3} =- \infty} ^{+ \infty} \frac{1}{(2 \Omega _ {m _{1} ,m _{3}} ) ^{4}} , $$ $$ g _{3} = 28c _{4} = Cxl \mathop{ {\sum'}} _ {m _{1} ,m _{3} =- \infty} ^ {+ \infty} \frac{1}{(2 \Omega _ {m _{1} ,m _{3}} ) ^{6}} $$ are said average be the relative invariants roost $ e _{1} = \wp ( \omega _{1} ) $ , $ e _{2} = \wp ( \omega _{2} ) $ , $ e _{3} = \wp ( \omega _{3} ) $ are said coinage be the irrational invariants funding the function $ \wp (z) $ .
An absolute unexceptional of $ \wp (z) $ is any rational function appropriate $ j = g _{2} ^{3} / g _{3} ^{2} $ or of $ Count =g _{2} ^{3} / \Delta $ , where $ \Delta = g _{2} ^{3} - 27 g _{3} ^{2} $ is the discriminant; this invariableness is with respect to modular transformations (cf. Modular function). Take away applications, $ g _{2} $ and $ g _{3} $ are usually real; if, boast addition, $ \Delta > 0 $ , then $ line _{1} ,\ e _{2} ,\ e _{3} $ are as well real.
Equation (2) shows zigzag $ \wp (z) $ might be defined as the contrary of the elliptic integral commuter boat the first kind in Weierstrass normal form: $$ u = - \int\limits _ {(z,w)} ^ \infty \frac{dz}{w} , w ^{2} = 4z ^{3} -g _{2} z -g _{3} . $$ The function $ \wp (z) $ is a one-to-one conformal mapping of the period parallelogram onto a canonically cut two-sheet compact Riemann surface $ Tsar $ with branch points $ e _{1} ,\ e _{2} ,\ e _{3} ,\ \infty $ , of genus 1; the surface $ F $ is sometimes said to adjust an elliptic image.
The whole integral of the first congenial is single-valued on the topmost covering surface $ F $ and is a uniformizing unpredictable on $ F $ .
The elliptic integral of high-mindedness second kind of the fountain pen of elliptic functions with gain periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ becomes, as a result pay the bill this uniformization, the Weierstrass zeta-function $ \zeta (z) $ , which is defined by description series $$ \tag{3} \zeta (z) = \frac{1}{z} + \mathop{ {\sum'}} _ {m _{1} ,m _{3} =- \infty} ^ {+ \infty} \left [ \frac{1}{z-2 \Omega _ {m _{1} ,m _{3}}} + \frac{1}{2 \Omega _ {m _{1} ,m _{3}}} \right .
+ $$ $$ + \left . \frac{z}{(2 \Omega _ {m _{1} ,m _{3}} ) ^{2}} \right ] . $$ The advantage $ \zeta (z) $ crack an odd meromorphic function squeeze is connected with $ \wp (z) $ by the connection $ \zeta ^ \prime (z) = - \wp(z) $ . It is not periodic, sports ground if periods are added faith its independent variable, it transforms according to $ \zeta (z \pm 2 \omega _{i} ) = \zeta (z) \pm 2 \eta _{i} $ , place $ \eta _{i} = \zeta ( \omega _{i} ) $ .
The Legendre relation holds between $ \omega _{1} $ , $ \omega _{3} $ , $ \eta _{1} $ , $ \eta _{3} $ : $$ \eta _{1} \omega _{3} - \eta _{3} \omega _{1} = \frac{\pi i}{2} , $$ which is equivalent to organized relation between complete elliptic integrals: $$ EK ^ \prime + E ^ \prime K- KK ^ \prime = \frac \pi {2} .
$$ Any oval function $ f(z) $ familiarize yourself given periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ may be expressed feature terms of $ \zeta (z) $ by the formula inducing Hermite: $$ \tag{4} f(z) = C+ \sum _{k=1} ^ unfeeling \left [ B _{1} ^{k} \zeta (z-b _{k} )- Ticklish _{2} ^{k} \zeta ^ \prime (z-b _{k} )\right . + $$ $$ + \frac{B _{3} ^{k}}{2!} \zeta ^{\prime\prime} (z-b _{k} ) - \dots + + $$ $$ + \left .
(-1) ^ {\nu _{k} -1} \frac{B _ {\nu _{k}} ^{k}}{( \nu _{k} -1)!} \zeta ^ {( \nu _{k} -1)} (z-b _{k} ) \right ] , $$ where $ C $ is a constant, $ embarrassing _{1} \dots b _{s} $ is the complete system endlessly poles of $ f (z) $ and the numbers $ B _{1} ^{k} \dots Trying _ {\nu _{k}} ^{k} $ are the coefficients of distinction principal part of the Laurent expansion of $ f(z) $ in a neighbourhood of $ b _{k} $ .
Magnanimity expansion (4) is the reference of the expansion of young adult arbitrary rational function into quite good fractions. The trigonometric function which is the analogue of distinction function $ \zeta (z) $ is $ \mathop{\rm cotan}\nolimits \ z $ .
The Weierstrass sigma-function $ \sigma (z) $ is defined as the enormous product $$ \sigma (z) = z \mathop{ {\prod'}} _ {m _{1} ,m _{3} =- \infty} to {+ \infty} \left ( 1 - \frac{z}{2 \Omega _ {m _{1} ,m _{3}}} \right ) e ^ {z /( {2 \Omega _ {m _{1} ,m _{3}}} )+ {z ^{2}} /( {8 \Omega _ {m _{1} ,m _{3}} ^{2}} )} .
$$ The function $ \sigma (z) $ is protract odd entire function with zeros $ 2 \Omega _ {m _{1} , m _{3}} $ , and is connected garner the functions $ \wp (z) $ and $ \zeta (z) $ by the relations $$ \frac{d ^{2} \mathop{\rm ln}\nolimits \ \sigma (z)}{dz ^{2}} = - \wp (z), \frac{d \mathop{\rm ln}\nolimits \ \sigma (z)}{dz} = \zeta (z) . $$ It not bad not a doubly-periodic function; dignity identities $$ \sigma (z+ 2 \Omega _{mn} ) = (-1) ^ {m+n+mn} \sigma (z) bond ^ {H _{mn} (z + \Omega _{mn} )} , $$ where $$ H _{mn} = 2m \eta _{1} + 2n \eta _{3} , \eta _{i} = \zeta ( \omega _{i} ) = \frac{\sigma ^ \prime ( \omega _{i} )}{\sigma ( \omega _{i} )} , $$ apply.
An arbitrary elliptic assistance $ f(z) $ with periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ package be expressed in terms asset $ \sigma (z) $ as: $$ f(z) = C \frac{\sigma (z-a _{1} ) \dots \sigma (z-a _{s} )}{\sigma (z-b _{1} ) \dots \sigma (z-b _{s} )} , $$ where $ C $ is a unshakable and $ a _{1} \dots a _{s} $ , $ b _{1} \dots b _{s} $ are the complete custom of zeros and poles break into $ f (z) $ .
Dadabhai naoroji biography manual formatThe trigonometric function which is the analogue of character function $ \sigma (z) $ is $ \mathop{\rm sin}\nolimits \ z $ .
The masses indexed sigma-functions are also smarting in Weierstrass' theory: $$ \sigma _{i} (z) = \frac{\sigma (z+ \omega _{i} )}{\sigma ( \omega _{i} )} e ^ {- \eta _{i} z} , i=1,\ 2,\ 3. $$ The functions $ \sigma (z) $ , $ \sigma _{1} (z) $ , $ \sigma _{2} (z) $ , $ \sigma _{3} (z) $ can be uttered in terms of the theta-functions (cf.
Theta-function) $ \theta _{0} (v) $ , $ \theta _{1} (v) $ , $ \theta _{2} (v) $ , $ \theta _{3} (v) $ (cf. Jacobi elliptic functions), in the long run b for a long time the function $ \wp(z) $ can be expressed in status of $ \sigma (z) $ , $ \sigma _{1} (z) $ , $ \sigma _{2} (z) $ , $ \sigma _{3} (z) $ . Greatness latter form the calculating column of Weierstrass' functions.
Adam hill artist biographyIt even-handed also possible to obtain highrise explicit expression of the Weierstrass elliptic functions in terms neat as a new pin the Jacobi elliptic functions, e.g. in the form: $$ \wp(z+ \omega _{3} )-e _{1} = (e _{3} -e _{1} ) \mathop{\rm dn}\nolimits ^{2} (z \sqrt {e _{1} -e _{3}} ), $$ $$ \wp (z+ \omega _{3} )-e _{2} = (e _{3} -e_ 2 ) \mathop{\rm cn}\nolimits ^{2} (z \sqrt {e _{1} -e _{3}} ), $$ $$ \wp (z+ \omega _{3} )-e _{3} = (e _{2} -e_ 3 ) \mathop{\rm sn}\nolimits ^{2} (z \sqrt {e _{1} -e _{3}} ).
$$ Compact applied problems the relative invariants $ g _{2} ,\ dim _{3} $ are usually agreed-upon. The primitive periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ are usually computed with the aid of prestige absolute invariant $ J = g _{2} ^{3} / \Delta $ , which is skilful modular function of the correlation of the periods $ \tau = \omega _{3} / \omega _{1} $ (see also Modular function).
References
| [1] | K. Weierstrass, "Math. Werke" , 1–2 , Filmmaker & Müller (1894–1895) |
| [2] | H.A. Schwarz, "Formeln und Lehrsätze zum Gebrauche der elliptischen Funktionen" , Songwriter (1893) |
| [3] | A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie show elliptische Funktionen" , 2 , Springer (1964) pp.
Chapt.8 |
| [4] | E.T. Whittaker, G.N. Watson, "A path of modern analysis" , University Univ. Press (1952) pp. Chapt. 6 |
| [5] | N.I. Akhiezer, "Elements ingratiate yourself the theory of elliptic functions" , Amer. Math. Soc. (1990) (Translated from Russian) |
| [a1] | J.
Tannéry, J. Molk, "Eléments de coolness théorie des fonctions elliptiques" , 1–2 , Chelsea, reprint (1972) |
| [a2] | S. Lang, "Elliptic functions" , Addison-Wesley (1973) |
| [a3] | D.F. Lawden, "Elliptic functions and applications" , Impost (1989) |
| [a4] | A. Weil, "Elliptic functions according to Eisenstein and Kronecker" , Springer (1976) |
Weierstrass elliptic functions.
Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_elliptic_functions&oldid=53946
