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ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi)

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ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi) Math 213a (Fall 2021) Yum-Tong Siu 1 ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi) Significance of Elliptic Functions. Elliptic functions and their associated theta functions are a new class of special functions which play an impor- tant role in explicit solutions of real world problems. Elliptic functions as meromorphic functions on compact Riemann surfaces of genus 1 and their associated theta functions as holomorphic sections of holomorphic line bun- dles on compact Riemann surfaces pave the way for the development of the theory of Riemann surfaces and higher-dimensional abelian varieties. Two Approaches to Elliptic Function Theory. One approach (which we call the approach of Abel and Jacobi) follows the historic development with motivation from real-world problems and techniques developed for solving the difficulties encountered. One starts with the inverse of an elliptic func- tion defined by an indefinite integral whose integrand is the reciprocal of the square root of a quartic polynomial. An obstacle is to show that the inverse function of the indefinite integral is a global meromorphic function on C with two R-linearly independent primitive periods. The resulting dou- bly periodic meromorphic functions are known as Jacobian elliptic functions, though Abel was actually the first mathematician who succeeded in inverting such an indefinite integral. Nowadays, with the use of the notion of a Rie- mann surface, the inversion can be handled by using the fundamental group of the Riemann surface constructed to make the square root of the quartic polynomial single-valued. The great advantage of this approach is that there is vast literature for the properties of the Jacobain elliptic functions and of their associated Jacobian theta functions. Moreover, the trigonometric functions are the special cases of the Jacobian elliptic functions when the (elliptic) modulus degenerates to zero so that the properties and relations of trigonometric functions can serve a guide, albeit an extremely crude one, to those of the Jacobian elliptic functions and Jacobian theta functions. The second approach, which we call the approach of Weierstrass, starts with the construction of an infinite sum of partial fraction expansions, known as the Weierstrass } function, and then verifies that the function it represents is a doubly periodic meromorphic function on C which satisfies a first-order differential equation so that it is the inverse of an indefinite integral whose integrand is the reciprocal of the square root of a cubic polynomial. The second approach of Weierstrass is Math 213a (Fall 2021) Yum-Tong Siu 2 that it is cleaner, easier, and more elegant to develop. Moreover, it fits in more naturally with the geometric theory of (complex) cubic plane curves. Both Approaches Are Treated in the Course. We will discuss both ap- proaches in this course, first the inversion of the inverse Jacobian elliptic sine function defined by an indefinite integral whose integrand is the reciprocal of the square root of a quartic polynomial. Properties and relations of the several Jacobian elliptic functions will be discussed. Then we will introduce the Weierstrass } function as an infinite sum of partial fractions and derive its properties and its relation with the Jacobian elliptic sine function. Then we will treat the Jacobian theta functions and their applications to the sums of squares problem and quadratic reciprocity. We start with the motion of the simple pendulum, with more details, for whose solution the Jacobian el- liptic sine function with (elliptic) modulus k is introduced as the inverse of the indefinite integral Z ξ 1 sn−1ξ = : p 2 2 2 x=0 (1 − x )(1 − k x ) Exact Solution of Motion of Simple Pendulum. Let m be the mass of the bob at the end of the pendulum, a be the length of the pendulum, θ be the angle of inclination which the pendulum makes with a vertical line, α be the initial angle of inclination when the pendulum is released from rest position at time zero, t be the time variable, and g be the constant of the gravity of the earth. The equation of the conservation of energy is 1 dθ2 ma2 − mga cos θ = −mga cos α: 2 dt It follows that dθ2 g g α θ = 2 (cos θ − cos α) = 4 sin2 − sin2 : dt a a 2 2 The motivation to use the double angle formula for the cosine function to get the last expression is that when one tries to simplify Z dx p(1 − x2) (1 − k2x2) Math 213a (Fall 2021) Yum-Tong Siu 3 p by using the substitution x = sin ' to get rid of 1 − x2, one gets the integral Z d' p 1 − k2 sin2 ' and in that integral there is the expression sin2 '. In order to get an integral equivalent to the form Z d' p ; 1 − k2 sin2 ' we use the substitution θ sin 2 sin ' = α : sin 2 One gets the differential equation d'2 g α = 1 − sin2 sin2 ' : dt a 2 So a Z ' d t = q : g =0 2 α 2 1 − sin 2 sin The final answer is θ α rg sin = sin sn t 2 2 a α with the (elliptic) modulus k equal to sin 2 . The periodic time of the pendulum is π ra ra Z 2 d' 4 K = 4 p 2 g g 0 1 − k2 sin ' π ra Z 2 1 1 · 3 = 4 1 + k2 sin2 ' + k4 sin4 ' + ··· d' g 0 2 2 · 4 ! ra 12 1 · 32 = 2π 1 + k2 + k4 + ··· ; g 2 2 · 4 because π Z 2 π 1 · 3 · 5 ··· (2n − 1) sin2n 'd' = · ; 0 2 2 · 4 · 6 ··· (2n) Math 213a (Fall 2021) Yum-Tong Siu 4 from π 2π 2π 2n Z 2 1 Z 1 Z 1 sin2n 'd' = sin2n 'd' = ei' − e−i' d' 0 4 0 4 0 2i 1 Z 2π 1 2n 2n 1 1 2n 2n = (−1)n d' = 2π (−1)n : 4 0 2i n 4 2i n The purpose of the formula about the periodic time of the pendulum is to show the order of the error involved when the trigonometric sine function is used instead of the Jacobian elliptic sine function. Inversion by Addition Formula by Abel. Abel in his 1827 paper used the addition formula to invert the indefinite integral Z ξ 1 sn−1ξ = : p 2 2 2 x=0 (1 − x )(1 − k x ) To understand the role of the additional theorem, let us consider first the simpler case of trigonometric functions (also known as circular functions). The inverse sine function is given by Z ξ dx θ = sin−1 ξ = p : 2 x=0 1 − x Because of monotonicity from the positivity of the derivative p 1 , we can 1−ξ2 π π always invert for the range of −1 ≤ ξ ≤ 1 to get the range of θ in − 2 ; 2 . We know that the function θ = sin−1 ξ of the indefinite integral is not single- valued. To invert it, we hope to get a single-valued function ξ = sin θ so that the multivaluedness of θ = sin−1 ξ would mean the periodicity of ξ = sin θ. For fixed −1 ≤ ξ ≤ 1 we would like to know, besides one value θ with ξ = sin θ, what other values θ0 would also give ξ = sin θ0. It is a matter of comparing sin θ with sin(θ + ') (where ' = θ0 − θ). For that reason, one would like to use the addition formula to express sin(θ + ') and sin(θ − ') in terms of sin θ, cos θ, sin ' and cos '. In 1827 Abel actually used this method to invert the indefinite integral which defines an elliptic function and was the first person to succeed the inversion. Math 213a (Fall 2021) Yum-Tong Siu 5 To understand how one can derive the addition formula from the indefinite integral of the inverse function, one first looks at the simpler case of the circular functions (i.e., the trigonometric functions). The addition formula is sin(θ + ') = sin θ cos ' + cos θ sin ' q p = sin θ 1 − sin2 ' + 1 − sin2 θ sin ': Let ξ = sin θ and η = sin ' so that θ = sin−1 ξ and ' = sin−1 η. The addition formula expressed in terms of ξ and η now becomes sin−1 ξ + sin−1 η = θ + ' q p = sin−1 sin θ 1 − sin2 ' + 1 − sin2 θ sin ' p p = sin−1 ξ 1 − η2 + 1 − ξ2η ; which, in terms of the defining indefinite integrals, is p p Z ξ dx Z η dx Z ξ 1−η2+ 1−ξ2η dx p + p = p : 2 2 2 x=0 1 − x x=0 1 − x x=0 1 − x The addition theorem tells us that the sum of two definite integrals is equal to a third definite integral whose upper limit is expressed in terms of the upper limits of the original two definite integrals. The upper term of each integral is the variable for the inverse function in the addition formula. Once we have such an equation of three definite integrals with appropriate upper limits, we can always check it by differentiating both sides with respect to the upper limit of the first integral and using the fundamental theorem of calculus. The challenge is to guess from the additional formula for the circular functions (as a special case of the elliptic functions) what the addition formula should be for the elliptic functions (if there is one at all). Abel in his 1827 paper introduced many ingenious procedures to get the additional formula for elliptic functions. After Riemann introduced the concept of a Riemann surface, nowadays the inversion process to define an elliptic function over all of C is done by using the fundamental group.
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