From trigonometry to elliptic functions
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1 From trigonometry to elliptic functions Zhiqin Lu The Math Club University of California, Irvine March 3, 200 Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions /24
2 Question What is the area of a triangle with side lengths a, b, and c? Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 2/24
3 Heron s formula [A.D. 60] A = p(p a)(p b)(p c), where p = (a + b + c)/2. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 3/24
4 By the Pythagorean theorem, we have b 2 d 2 = a 2 (c d) 2. Therefore, we have d = (c + b2 a 2 ). 2 c By the Pythagorean theorem again, we have h 2 = b 2 (c + b2 a 2 ) 2. 4 c The formula follows from the fact that Figure: Triangle with altitude h cutting base c into d and (c - d). A = 2 ch. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 4/24
5 Using Trigonometry, the formula is a lot easier to prove: A = bc sin α. 2 By the law of cosine, we have sin α = ( ) b2 cos 2 + c α = 2 a bc Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 5/24
6 What is sin 20? Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 6/24
7 What is sin 20? 2 We have to use the concept function. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 6/24
8 What is sin 20? 2 We have to use the concept function. 3 A function is an assignment: f : A B. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 6/24
9 What is sin 20? 2 We have to use the concept function. 3 A function is an assignment: f : A B. 4 How to define trigonometric functions? Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 6/24
10 Definition: sin A = a/h cos A = b/h tan A = a/b Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 7/24
11 Definition: sin A = a/h cos A = b/h tan A = a/b Such a definition can hardly be used in Calculus! Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 7/24
12 How to find the derivative of sin x? We have to use/assume the fact which is the derivative of sin x at 0. sin θ lim =, θ 0 θ Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 8/24
13 In Stewart s or Minton/Smith s Calculus book, there are two ways to define the exp/log functions Define e x first, and log x is the inverse function of e x Define e x for x integers; 2 Define e x for x rational numbers; 3 Define e x for any real number x by continuity. 2 Define log x first and then define e x as the inverse of log x log x = dx + C. x Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 9/24
14 Can we do that same thing for trigonometric functions? Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 0/24
15 Can we do that same thing for trigonometric functions? arcsin x = dx + C x 2 Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 0/24
16 Can we do that same thing for trigonometric functions? arcsin x = dx + C x 2 Compare with the integral dx = log(x + + x 2 ) + C + x 2 We get the following mysterious formula log(x + + x 2 ) = arcsin( x) Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 0/24
17 A complex number z is a pair of two real numbers. But we are more used to writing it as z = x + iy = x + y, where x, y are the two real numbers. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions /24
18 A complex number z is a pair of two real numbers. But we are more used to writing it as z = x + iy = x + y, where x, y are the two real numbers. The production of two complex numbers is somewhat interesting (x + iy )(x 2 + iy 2 ) = x x 2 y y 2 + i(x y 2 + x 2 y ). Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions /24
19 A complex number z is a pair of two real numbers. But we are more used to writing it as z = x + iy = x + y, where x, y are the two real numbers. The production of two complex numbers is somewhat interesting (x + iy )(x 2 + iy 2 ) = x x 2 y y 2 + i(x y 2 + x 2 y ). The negative sign costs 99% of errors! Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions /24
20 Euler s formula e iz = cos z + i sin z Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 2/24
21 Euler s formula e iz = cos z + i sin z cos z = eiz + e iz 2 sin z = eiz e iz 2i Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 2/24
22 Theorem: cos(x + y) = cos x cos y sin x sin y. Proof. Q.E.D. cos x cos y sin x sin y = eix + e ix 2 eiy + e iy 2 eix e ix 2i = 4 (ei(x+y) + e i(x+y) + e i(x y) + e i(y x) ) eiy e iy 2i + 4 (ei(x+y) + e i(x+y) e i(x y) e i(y x) ) = cos(x + y). Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 3/24
23 Theorem: cos(x + y) = cos x cos y sin x sin y. Proof. cos x cos y sin x sin y = eix + e ix 2 eiy + e iy 2 eix e ix 2i = 4 (ei(x+y) + e i(x+y) + e i(x y) + e i(y x) ) eiy e iy 2i + 4 (ei(x+y) + e i(x+y) e i(x y) e i(y x) ) = cos(x + y). Q.E.D. (=quod erat demonstrandum Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 3/24
24 Theorem: cos(x + y) = cos x cos y sin x sin y. Proof. cos x cos y sin x sin y = eix + e ix 2 eiy + e iy 2 eix e ix 2i = 4 (ei(x+y) + e i(x+y) + e i(x y) + e i(y x) ) eiy e iy 2i + 4 (ei(x+y) + e i(x+y) e i(x y) e i(y x) ) = cos(x + y). Q.E.D. (=quod erat demonstrandum=that which was to be demonstrated) Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 3/24
25 The above discussion hint us that we need to study dz, z 2 where z is a complex number. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 4/24
26 The above discussion hint us that we need to study dz, z 2 where z is a complex number. Or more precisely, φ(z) = z 0 t 2 dt. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 4/24
27 The function z 2 is multi-valued. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 5/24
28 The function z 2 is multi-valued. 2 In order to make it single-valued, we need to construct a Riemann surface. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 5/24
29 The function z 2 is multi-valued. 2 In order to make it single-valued, we need to construct a Riemann surface. 3 We cut two copies of C along [, ] and glue them. The space C is called a Riemann surface. z 2 is a single-valued function on the Riemann surface C. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 5/24
30 The function z 2 is multi-valued. 2 In order to make it single-valued, we need to construct a Riemann surface. 3 We cut two copies of C along [, ] and glue them. The space C is called a Riemann surface. z 2 is a single-valued function on the Riemann surface C. 4 φ(z) is multivalued even if on C, z 2 is single-valued, Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 5/24
31 The function z 2 is multi-valued. 2 In order to make it single-valued, we need to construct a Riemann surface. 3 We cut two copies of C along [, ] and glue them. The space C is called a Riemann surface. z 2 is a single-valued function on the Riemann surface C. 4 φ(z) is multivalued even if on C, z 2 5 because of the existence of Residue. is single-valued, Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 5/24
32 The function z 2 is multi-valued. 2 In order to make it single-valued, we need to construct a Riemann surface. 3 We cut two copies of C along [, ] and glue them. The space C is called a Riemann surface. z 2 is a single-valued function on the Riemann surface C. 4 φ(z) is multivalued even if on C, z 2 5 because of the existence of Residue. 6 We have z =R dz z 2 = 2π is single-valued, Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 5/24
33 Therefore, φ : C C/2πZ Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 6/24
34 Therefore, φ : C C/2πZ 2 φ : C/2πZ C exists. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 6/24
35 Therefore, φ : C C/2πZ 2 φ : C/2πZ C exists. 3 C is the set of all points (t, ± t 2 ). Therefore, C can be represented by the set x 2 + y 2 = in C 2. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 6/24
36 Therefore, φ : C C/2πZ 2 φ : C/2πZ C exists. 3 C is the set of all points (t, ± t 2 ). Therefore, C can be represented by the set x 2 + y 2 = in C 2. 4 C is a group (x, y ) (x 2, y 2 ) = (x y 2 + x 2 y, x x 2 y y 2 ). Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 6/24
37 Therefore, φ : C C/2πZ 2 φ : C/2πZ C exists. 3 C is the set of all points (t, ± t 2 ). Therefore, C can be represented by the set x 2 + y 2 = in C 2. 4 C is a group (x, y ) (x 2, y 2 ) = (x y 2 + x 2 y, x x 2 y y 2 ). Check: (x y 2 + x 2 y ) 2 + (x x 2 y y 2 ) 2 =. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 6/24
38 The map φ : C/2πZ C is a group isomorphism: (α, β) α + β, ((x, y ), (x 2, y 2 )) (x x 2 y y 2, x y 2 + x 2 y ) where x = cos α, y = sin α, etc. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 7/24
39 The map φ : C/2πZ C is a group isomorphism: (α, β) α + β, ((x, y ), (x 2, y 2 )) (x x 2 y y 2, x y 2 + x 2 y ) where x = cos α, y = sin α, etc. Such an isomorphism is called Addition Theorem. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 7/24
40 The map φ : C/2πZ C is a group isomorphism: (α, β) α + β, ((x, y ), (x 2, y 2 )) (x x 2 y y 2, x y 2 + x 2 y ) where x = cos α, y = sin α, etc. Such an isomorphism is called Addition Theorem. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 7/24
41 The following is another version of the addition theorem for the sine function: sin u 0 sin v dx + x 2 0 sin(u+v) dx = x 2 0 x 2 dx Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 8/24
42 The following is another version of the addition theorem for the sine function: sin u 0 sin v dx + x 2 If we set sin u, z = sin v, then we have 0 sin(u+v) dx = x 2 0 x 2 dx where y 0 z dx + x 2 0 T (y,z) dx = x 2 T (y, z) = y z 2 + z y 2. 0 x 2 dx, Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 8/24
43 The following is another version of the addition theorem for the sine function: sin u 0 sin v dx + x 2 If we set sin u, z = sin v, then we have 0 sin(u+v) dx = x 2 0 x 2 dx where y 0 z dx + x 2 0 T (y,z) dx = x 2 T (y, z) = y z 2 + z y 2. 0 x 2 dx, Any other functions satisfy the above addition theorem? Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 8/24
44 The following is another version of the addition theorem for the sine function: sin u 0 sin v dx + x 2 If we set sin u, z = sin v, then we have 0 sin(u+v) dx = x 2 0 x 2 dx where y 0 z dx + x 2 0 T (y,z) dx = x 2 T (y, z) = y z 2 + z y 2. 0 x 2 dx, Any other functions satisfy the above addition theorem? Elliptic Functions! Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 8/24
45 Consider the integral dt t(t )(t λ), Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 9/24
46 Consider the integral dt t(t )(t λ), where 0 < λ <. Historic Remarks Legendre studied these kind of integral extensively and wrote 3 volumes (4 volumes?) of Traite des fonctions elliptiques. But he failed in finding the most important properties of the integrals: they are the inverse functions of some doubly periodic functions! This fact was found by Abel and Jocobi independently. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 9/24
47 The Riemann surface of the function Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 20/24
48 The Riemann surface of the function z(z )(z λ) Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 20/24
49 The Riemann surface of the function z(z )(z λ) y 2 = z(z )(z λ) Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 20/24
50 Let ω = ω 2 = A B dz z(z )(z λ) dz z(z )(z λ) Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 2/24
51 Let ω = ω 2 = A B dz z(z )(z λ) dz z(z )(z λ) Then the inverse function of the elliptic integral z dt t(t )(t λ) is a doubly period function of periods ω, ω 2. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 2/24
52 Basic property of elliptic functions Theorem: Entire (holomorphic) doubly periodic functions are constants. Proof. Bounded holomorphic functions are constant by Liouivile Theorem. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 22/24
53 Basic property of elliptic functions Theorem: Entire (holomorphic) doubly periodic functions are constants. Proof. Bounded holomorphic functions are constant by Liouivile Theorem. Therefore, elliptic functions are meromorphic Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 22/24
54 Basic property of elliptic functions Theorem: Entire (holomorphic) doubly periodic functions are constants. Proof. Bounded holomorphic functions are constant by Liouivile Theorem. Therefore, elliptic functions are meromorphic (singularities are poles). Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 22/24
55 Theorem Any two tori T (ω, ω 2 ) and T (ω 3, ω 4 ) are biholomorphic if and only if there exists integers a, b, c, d such that ad bc =, and ω 4 ω 3 = aω + bω 2 cω + dω 2. Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 23/24
56 References: n2/paper2/v0n2-2pd.pdf 2 Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions 24/24
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