Weierstrass elliptic functions

Functions on which K. Weierstrass based his general theory of elliptic functions (cf. Elliptic function), exposed in 1862 in his lectures at the University of Berlin [1], [2]. As distinct from the earlier structure of the theory of elliptic functions developed by A. Legendre, N.H. Abel and C.G. Jacobi, which was based on elliptic functions of the second order with two simple poles in the period parallelogram, a Weierstrass elliptic function has one second-order pole in the period parallelogram. From the theoretical point of view the theory of Weierstrass is simpler, since the function $ {\mathbf p} (z) $ , on which it is based, and its derivative serve as elliptic functions which generate the algebraic field of elliptic functions with given primitive periods.

The Weierstrass $ {\mathbf p} $ -function $ {\mathbf p} (z) $ ( $ {\mathbf p} $ is Weierstrass' notation) for given primitive periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ , $ \mathop{\rm Im}\nolimits ( \omega _{3} / \omega _{1} ) > 0 $ , is defined as the series $$ \tag{1} {\mathbf p} (z) = {\mathbf p} (z; \ 2 \omega _{1} ,\ 2 \omega _{3} ) = $$ $$ = \frac{1}{z ^{2}} + \mathop{ {\sum'}} _ {m _{1} , m _{3} = - \infty } ^ {+ \infty} \left [ \frac{1}{(z-2 \Omega _ {m _{1} , m _{3}} ) ^{2} } - \frac{1}{(2 \Omega _ {m _{1} ,m _{3}} ) ^{2}} \right ] = $$ $$ = \frac{1}{z ^{2}} + c _{2} z ^{2} + c _{4} z ^{4} + \dots , $$ where $ \Omega _ {m _{1} , m _{3}} = m _{1} \omega _{1} +m _{3} \omega _{3} $ , and $ m _{1} ,\ m _{3} $ run through all integers except $ m _{1} = m _{3} = 0 $ . The function $ {\mathbf p} (z) $ is an even elliptic function of order 2, with a unique second-order pole with zero residue in each period parallelogram. Its derivative $ {\mathbf p} ^ \prime (z) $ is an odd elliptic function of order 3 with the same primitive periods; $ {\mathbf p} ^ \prime (z) $ has simple zeros at points congruent with $ \omega _{1} ,\ \omega _{2} = \omega _{1} + \omega _{3} ,\ \omega _{3} $ . The most important property of the function $ {\mathbf p} (z) $ is that any elliptic function with given primitive periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ may be represented as a rational function of $ {\mathbf p} (z) $ and $ {\mathbf p} ^ \prime (z) $ , i.e. $ {\mathbf p} (z) $ and $ {\mathbf p} ^ \prime (z) $ generate the algebraic field of elliptic functions with given periods. The simply-periodic trigonometric function which serves as the analogue of the function $ {\mathbf p} (z) $ is $ 1/ \mathop{\rm sin}\nolimits ^{2} \ z $ .

The function $ {\mathbf p} (z) $ satisfies the differential equation $$ \tag{2} {\mathbf p} ^ \prime2 (z) = 4 {\mathbf p} ^{3} (z)- g _{2} {\mathbf p} (z) -g _{3 } \equiv $$ $$ \equiv 4 [ {\mathbf p} (z) -e _{1} ] [ {\mathbf p} (z)-e _{2} ] [ {\mathbf p} (z) -e _{3} ], e _{1} +e _{2} +e _{3} = 0, $$ in which the modular forms $$ g _{2} = 20 c _{2} = 60 \mathop{ {\sum'}} _ {m _{1} ,m _{3} =- \infty} to {+ \infty} \frac{1}{(2 \Omega _ {m _{1} ,m _{3}} ) ^{4}} , $$ $$ g _{3} = 28c _{4} = 140 \mathop{ {\sum'}} _ {m _{1} ,m _{3} =- \infty} ^ {+ \infty} \frac{1}{(2 \Omega _ {m _{1} ,m _{3}} ) ^{6}} $$ are said to be the relative invariants and $ e _{1} = {\mathbf p} ( \omega _{1} ) $ , $ e _{2} = {\mathbf p} ( \omega _{2} ) $ , $ e _{3} = {\mathbf p} ( \omega _{3} ) $ are said to be the irrational invariants of the function $ {\mathbf p} (z) $ . An absolute invariant of $ {\mathbf p} (z) $ is any rational function of $ j = g _{2} ^{3} / g _{3} ^{2} $ or of $ J =g _{2} ^{3} / \Delta $ , where $ \Delta = g _{2} ^{3} - 27 g _{3} ^{2} $ is the discriminant; this invariance is with respect to modular transformations (cf. Modular function). In applications, $ g _{2} $ and $ g _{3} $ are usually real; if, in addition, $ \Delta > 0 $ , then $ e _{1} ,\ e _{2} ,\ e _{3} $ are also real. Equation (2) shows that $ {\mathbf p}(z) $ may be defined as the inverse of the elliptic integral of 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 $ {\mathbf p} (z) $ is a one-to-one conformal mapping of the period parallelogram onto a canonically cut two-sheet compact Riemann surface $ F $ with branch points $ e _{1} ,\ e _{2} ,\ e _{3} ,\ \infty $ , of genus 1; the surface $ F $ is sometimes said to be an elliptic image. The above integral of the first kind is single-valued on the principal covering surface $ F $ and is a uniformizing variable on $ F $ .

The elliptic integral of the second kind of the field of elliptic functions with given periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ becomes, as a result of this uniformization, the Weierstrass zeta-function $ \zeta (z) $ , which is defined by the 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 function $ \zeta (z) $ is an odd meromorphic function and is connected with $ {\mathbf p} (z) $ by the relation $ \zeta ^ \prime (z) = - {\mathbf p} (z) $ . It is not periodic, and if periods are added to its independent variable, it transforms according to $ \zeta (z \pm 2 \omega _{i} ) = \zeta (z) \pm 2 \eta _{i} $ , where $ \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 a relation between complete elliptic integrals: $$ EK ^ \prime + E ^ \prime K- KK ^ \prime = \frac \pi {2} . $$ Any elliptic function $ f(z) $ with given periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ may be expressed in terms of $ \zeta (z) $ by the formula of Hermite: $$ \tag{4} f(z) = C+ \sum _{k=1} ^ s \left [ B _{1} ^{k} \zeta (z-b _{k} )- B _{2} ^{k} \zeta ^ \prime (z-b _{k} )\right . + $$ $$ + \frac{B _{3} ^{k}}{2!} \zeta ^{\prime\prime} (z-b _{k} ) - \dots + + $$ $$ + \left . (-1) ^ {\nu _{k} -1} \frac{B _ {\nu _{k}} ^{k}}{( \nu _{k} -1)!} \zeta ^ {( \nu _{k} -1)} (z-b _{k} ) \right ] , $$ where $ C $ is a constant, $ b _{1} \dots b _{s} $ is the complete system of poles of $ f (z) $ and the numbers $ B _{1} ^{k} \dots B _ {\nu _{k}} ^{k} $ are the coefficients of the principal part of the Laurent expansion of $ f(z) $ in a neighbourhood of $ b _{k} $ . The expansion (4) is the analogue of the expansion of an arbitrary rational function into partial fractions. The trigonometric function which is the analogue of the function $ \zeta (z) $ is $ \mathop{\rm cotan}\nolimits \ z $ .

The Weierstrass sigma-function $ \sigma (z) $ is defined as the infinite 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 an odd entire function with zeros $ 2 \Omega _ {m _{1} , m _{3}} $ , and is connected with the functions $ {\mathbf p} (z) $ and $ \zeta (z) $ by the relations $$ \frac{d ^{2} \mathop{\rm ln}\nolimits \ \sigma (z)}{dz ^{2}} = - {\mathbf p} (z), \frac{d \mathop{\rm ln}\nolimits \ \sigma (z)}{dz} = \zeta (z) . $$ It is not a doubly-periodic function; the identities $$ \sigma (z+ 2 \Omega _{mn} ) = (-1) ^ {m+n+mn} \sigma (z) e ^ {H _{mn} (z + \Omega _{mn} )} , $$ 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 function $ f(z) $ with periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ can be expressed in terms of $ \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 constant and $ a _{1} \dots a _{s} $ , $ b _{1} \dots b _{s} $ are the complete system of zeros and poles of $ f (z) $ . The trigonometric function which is the analogue of the function $ \sigma (z) $ is $ \mathop{\rm sin}\nolimits \ z $ .

The following indexed sigma-functions are also important 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 expressed 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), while the function $ {\mathbf p} (z) $ can be expressed in terms of $ \sigma (z) $ , $ \sigma _{1} (z) $ , $ \sigma _{2} (z) $ , $ \sigma _{3} (z) $ . The latter form the calculating base of Weierstrass' functions. It is also possible to obtain an explicit expression of the Weierstrass elliptic functions in terms of the Jacobi elliptic functions, e.g. in the form: $$ {\mathbf p} (z+ \omega _{3} )-e _{1} = (e _{3} -e _{1} ) \mathop{\rm dn}\nolimits ^{2} (z \sqrt {e _{1} -e _{3}} ), $$ $$ {\mathbf p} (z+ \omega _{3} )-e _{2} = (e _{3} -e sub 2 ) \mathop{\rm cn}\nolimits ^{2} (z \sqrt {e _{1} -e _{3}} ), $$ $$ {\mathbf p} (z+ \omega _{3} )-e _{3} = (e _{2} -e sub 3 ) \mathop{\rm sn}\nolimits ^{2} (z \sqrt {e _{1} -e _{3}} ). $$ In applied problems the relative invariants $ g _{2} ,\ g _{3} $ are usually given. The primitive periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ are usually computed with the aid of the absolute invariant $ J = g _{2} ^{3} / \Delta $ , which is a modular function of the ratio of the periods $ \tau = \omega _{3} / \omega _{1} $ (see also Modular function).

References

[1]	K. Weierstrass, "Math. Werke" , 1–2 , Mayer & Müller (1894–1895)
[2]	H.A. Schwarz, "Formeln und Lehrsätze zum Gebrauche der elliptischen Funktionen" , Berlin (1893)
[3]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 2 , Springer (1964) pp. Chapt.8
[4]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6
[5]	N.I. Akhiezer, "Elements of the theory of elliptic functions" , Amer. Math. Soc. (1990) (Translated from Russian)

References

[a1]	J. Tannéry, J. Molk, "Eléments de la 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" , Springer (1989)
[a4]	A. Weil, "Elliptic functions according to Eisenstein and Kronecker" , Springer (1976)