Complex Numbers and Euler s Identity

Transcription

1 Complex Numbers and Euler s Identity MATH 171 Freshman Seminar for Mathematics Majors J. Robert Buchanan Department of Mathematics 2010

2 Background Easy: solve the equation 0 = 1 z 2. 0 = 1 z 2 = (1 z)(1 + z) 1 = z or 1 = z

3 Background Easy: solve the equation 0 = 1 z 2. 0 = 1 z 2 = (1 z)(1 + z) 1 = z or 1 = z Not (as) easy: solve the equation 0 = 1 + z 2. 0 = 1 + z 2 = 1 z 2 = ( 1 z)( 1 + z) 1 = z or 1 = z Define i = 1, then z = ±i.

4 Complex Numbers Definition A number of the form z = a + bi where a and b are real numbers and i = 1 is called a complex number. If a = 0 and b 0 so that z = bi, then z is called an imaginary number.

5 Picturing Complex Numbers z = a + bi i y a,b x

6 Polar Representation (1 of 2) z = re iθ i y a,b r Θ x

7 Polar Representation (2 of 2) z = a + bi = re iθ where ) θ = tan 1 ( b a and r = { a 2 + b 2 if a > 0, a 2 + b 2 if a < 0.

8 Examples (1 of 3) Example Find the polar representation of the following complex numbers. 1 + i 2 i 3 i

9 Examples (2 of 3) i y 1,1 x 3, 1 2, 1

10 Examples (3 of 3) Example Find the polar representation of the following complex numbers. 1 + i = 2e iπ/4 2 i = 5e i tan 1 ( 1/2) 5e i 3 i = 10e i tan 1 (1/3) 10e i

11 Complex Arithmetic If z 1 = a + bi and z 2 = c + di, then provided c 2 + d 2 0. z 1 + z 2 = (a + c) + (b + d)i z 1 z 2 = (a c) + (b d)i z 1 z 2 = (ac bd) + (ad + bc)i z 1 (ac + bd) + (bc ad)i = z 2 c 2 + d 2,

12 Complex Addition (1 of 3) ( 3 + 4i) + (5 2i) = 2 + 2i i y 3,4 z 1 x z 2 5, 2

13 Complex Addition (2 of 3) ( 3 + 4i) + (5 2i) = 2 + 2i i y 3,4 z 1 2,2 z 1 z 2 x z 2 5, 2

14 Complex Addition (3 of 3) ( 3 + 4i) + (5 2i) = 2 + 2i i y 3,4 z 2 z 1 2,2 z 1 z 2 z 1 x z 2 5, 2

15 Complex Multiplication z 1 z 2 = r 1 e iθ 1 r 2 e iθ 2 = (r 1 r 2 )e i(θ 1+θ 2 ) i y r2 Θ 2 r 1 r 1 r 2 Θ 1 Θ 2 Θ 1 x

16 Challenge Represent z = 1 = 1 + 0i in polar form.

17 Challenge Represent z = 1 = 1 + 0i in polar form. Since r = ( 1) = 1, then 1 = e iπ.

18 Challenge Represent z = 1 = 1 + 0i in polar form. Since r = ( 1) = 1, then 1 = e iπ. Rearranging the equation above yields an equation relating five of the most important constants in mathematics. e iπ + 1 = 0

19 Euler s Identity e iθ = cos θ + i sin θ i y r e iθ r cos Θ i sin Θ r Θ x

20 Commemorative Stamp

21 Complex Exponentiation Use Euler s Identity (e iθ = cos θ + i sin θ) to express z = i in polar form, and

22 Complex Exponentiation Use Euler s Identity (e iθ = cos θ + i sin θ) to express z = i in polar form, and evaluate i i.

23 Infinite Series In Calculus II you will learn to express the function e x as the infinite series: where n! = (1)(2)(3) (n). e x = 1 + x 1! + x 2 2! + x 3 3! +

24 Infinite Series In Calculus II you will learn to express the function e x as the infinite series: where n! = (1)(2)(3) (n). e x = 1 + x 1! + x 2 2! + x 3 3! + This infinite series holds for real and complex values of x.

25 Infinite Series In Calculus II you will learn to express the function e x as the infinite series: where n! = (1)(2)(3) (n). e x = 1 + x 1! + x 2 2! + x 3 3! + This infinite series holds for real and complex values of x. For example, 1 = e iπ = 1 + iπ 1! + (iπ)2 + (iπ)3 + 2! 3!

26 Expressing e iπ as a Series 1 = 1 + iπ 1! + (iπ)2 + (iπ)3 + (iπ)4 + (iπ)5 + 2! 3! 4! 5! = 1 + iπ π2 2 i π3 6 + π i π 5 120

27 Expressing e iπ as a Series 1 = 1 + iπ 1! + (iπ)2 + (iπ)3 + (iπ)4 + (iπ)5 + 2! 3! 4! 5! = 1 + iπ π2 2 i π3 6 + π i π Note: the series consists of alternating real and purely imaginary terms.

28 Expressing the Series for e iπ as a Graph 1 = 1 + iπ π2 2 i π3 6 + π i π i y x 1 2

29 Quadratic Map Suppose f(z) = z 2 + c where z and c can be complex numbers. Similar to the Newton s Method formula we may iterate the quadratic function f(z). Starting with z 0 we define for n = 1, 2,... z n = f(z n 1 ) = z 2 n 1 + c

30 Quadratic Map Suppose f(z) = z 2 + c where z and c can be complex numbers. Similar to the Newton s Method formula we may iterate the quadratic function f(z). Starting with z 0 we define for n = 1, 2,... z n = f(z n 1 ) = z 2 n 1 + c For example, if z 0 = 0 and c = 1 2 i then z 1 = 1 2 i z 2 z 3 = i = i.

31 Quadratic Map (Graph) i y x

32 General Exponentiation Suppose x and y are two real numbers and suppose y > x > 0. Question: which is larger x y or y x?

33 General Exponentiation Suppose x and y are two real numbers and suppose y > x > 0. Question: which is larger x y or y x? Remember one of our principles of mathematical inquiry, try some examples in order to gain insight into a complicated question. Let x = 2 and try y = 3, y = 4, and y = 5.

34 Equality (1 of 3) Since for some choices of 0 < x < y, x y > y x while for others x y < y x we may be curious about when x y = y x.

35 Equality (2 of 3) x y = y x ln(x y ) = ln(y x ) y ln x = x ln y Since y > x > 0 then y = kx where k > 1. Substitute this into the last equation above and solve for x and y in terms of k.

36 Equality (3 of 3) For k > 1, x = k 1/(k 1) y = k k/(k 1) y x

37 Limits as k 1 + Use l Hôpital s Rule to find lim x = lim k 1/(k 1) k 1 + k 1 + lim y = lim k k/(k 1). k 1 + k 1 +

38 Limits as k 1 + Use l Hôpital s Rule to find lim x = lim k 1/(k 1) k 1 + k 1 + lim y = lim k k/(k 1). k 1 + k 1 + lim k 1/(k 1) k 1 + lim k k/(k 1) k 1 + = e = e

39 Summary 6 5 x y y x 4 x y y x x y y 3 y x x y e,e 2 1 empty y x x

40 Students and Complex Numbers Student c Student c Bacchi i j 8 Bongiovanni i j 8 Cilladi i j 8 Cox 1 + i j 8 Crider i j 8 de Kok i j 8 Hansford i j 8 Hild i j 8 Junkin i j 8 Keglovits i j 8 Kibler i j 8 Konowal i j 8 Leber i j 8 Longo i j 8 Mecutchen i j 8 Miller, B i j 8 Miller, S i j 8 Nguyen i j 8 Reed 1 + i j 8 Smeltz i j 8 Starner i j 8 Visek i j 8 Williard i j 8 Zipko i j 8

41 Results Name: c = Results:

42 Homework Referring to the polygonal spiral approaching 1 on slide 28, find the total length of the spiral. For the complex number c you have been assigned and starting with z 0 = 0 iterate the quadratic map f(z) = z 2 + c ten times (or less if r = a 2 + b 2 > 2) for j = 1, 2,...,24. If all the iterates of the quadratic map have a magnitude of less than 2 record a result of 1, else record a result of 0.