Lecture 4/12: Polar Form and Euler s Formula. 25 Jan 2007

Transcription

1 Lecture 4/12: Polar Form and Euler s Formula MA154: Algebra for 1st Year IT Niall Madden Niall.Madden@NUIGalway.ie 25 Jan 2007 CS457 Lecture 4/12: Polar Form and Euler s Formula 1/17

2 Outline 1 Recall... Polar coordinates 2 Euler s formula. A little History cos x =... 3 Inverse, Products, and Quotients 4 Binomial Coefficients. 5 Some combinatorics 6 De Moivre s Theorem 6 De Moivre s Theorem 6 Applications of de Moirve s Theorem: CS457 Lecture 4/12: Polar Form and Euler s Formula 2/17

3 Recall... Polar coordinates Given z C then r = z and the angle θ that the line from the origin (0, 0) to the complex number z makes with the positive x-axis are called the Polar Coordinates of z. z = (r cos θ, r sin θ) = r (cos θ + i sin θ) is called the polar form of the complex number z = x + i y. Define arg z = {θ : z = r (cos θ + i sin θ)} for z 0. Any θ arg z is called an argument of z. If θ 1, θ 2 arg z then θ 2 = θ 1 + 2nπ for some integer n Z. The principal value Arg z of arg z is defined as the unique value θ arg z with π < θ π. So arg z = {Arg z + 2nπ : n Z}. Arg z = π z is a negative real number. CS457 Lecture 4/12: Polar Form and Euler s Formula 3/17

4 Recall... Polar coordinates Given z C then r = z and the angle θ that the line from the origin (0, 0) to the complex number z makes with the positive x-axis are called the Polar Coordinates of z. z = (r cos θ, r sin θ) = r (cos θ + i sin θ) is called the polar form of the complex number z = x + i y. Define arg z = {θ : z = r (cos θ + i sin θ)} for z 0. Any θ arg z is called an argument of z. If θ 1, θ 2 arg z then θ 2 = θ 1 + 2nπ for some integer n Z. The principal value Arg z of arg z is defined as the unique value θ arg z with π < θ π. So arg z = {Arg z + 2nπ : n Z}. Arg z = π z is a negative real number. CS457 Lecture 4/12: Polar Form and Euler s Formula 3/17

5 Recall... Polar coordinates Given z C then r = z and the angle θ that the line from the origin (0, 0) to the complex number z makes with the positive x-axis are called the Polar Coordinates of z. z = (r cos θ, r sin θ) = r (cos θ + i sin θ) is called the polar form of the complex number z = x + i y. Define arg z = {θ : z = r (cos θ + i sin θ)} for z 0. Any θ arg z is called an argument of z. If θ 1, θ 2 arg z then θ 2 = θ 1 + 2nπ for some integer n Z. The principal value Arg z of arg z is defined as the unique value θ arg z with π < θ π. So arg z = {Arg z + 2nπ : n Z}. Arg z = π z is a negative real number. CS457 Lecture 4/12: Polar Form and Euler s Formula 3/17

6 Recall... Polar coordinates Given z C then r = z and the angle θ that the line from the origin (0, 0) to the complex number z makes with the positive x-axis are called the Polar Coordinates of z. z = (r cos θ, r sin θ) = r (cos θ + i sin θ) is called the polar form of the complex number z = x + i y. Define arg z = {θ : z = r (cos θ + i sin θ)} for z 0. Any θ arg z is called an argument of z. If θ 1, θ 2 arg z then θ 2 = θ 1 + 2nπ for some integer n Z. The principal value Arg z of arg z is defined as the unique value θ arg z with π < θ π. So arg z = {Arg z + 2nπ : n Z}. Arg z = π z is a negative real number. CS457 Lecture 4/12: Polar Form and Euler s Formula 3/17

7 Recall... Polar coordinates Given z C then r = z and the angle θ that the line from the origin (0, 0) to the complex number z makes with the positive x-axis are called the Polar Coordinates of z. z = (r cos θ, r sin θ) = r (cos θ + i sin θ) is called the polar form of the complex number z = x + i y. Define arg z = {θ : z = r (cos θ + i sin θ)} for z 0. Any θ arg z is called an argument of z. If θ 1, θ 2 arg z then θ 2 = θ 1 + 2nπ for some integer n Z. The principal value Arg z of arg z is defined as the unique value θ arg z with π < θ π. So arg z = {Arg z + 2nπ : n Z}. Arg z = π z is a negative real number. CS457 Lecture 4/12: Polar Form and Euler s Formula 3/17

8 Recall... Polar coordinates Given z C then r = z and the angle θ that the line from the origin (0, 0) to the complex number z makes with the positive x-axis are called the Polar Coordinates of z. z = (r cos θ, r sin θ) = r (cos θ + i sin θ) is called the polar form of the complex number z = x + i y. Define arg z = {θ : z = r (cos θ + i sin θ)} for z 0. Any θ arg z is called an argument of z. If θ 1, θ 2 arg z then θ 2 = θ 1 + 2nπ for some integer n Z. The principal value Arg z of arg z is defined as the unique value θ arg z with π < θ π. So arg z = {Arg z + 2nπ : n Z}. Arg z = π z is a negative real number. CS457 Lecture 4/12: Polar Form and Euler s Formula 3/17

9 Recall... Polar coordinates Given z C then r = z and the angle θ that the line from the origin (0, 0) to the complex number z makes with the positive x-axis are called the Polar Coordinates of z. z = (r cos θ, r sin θ) = r (cos θ + i sin θ) is called the polar form of the complex number z = x + i y. Define arg z = {θ : z = r (cos θ + i sin θ)} for z 0. Any θ arg z is called an argument of z. If θ 1, θ 2 arg z then θ 2 = θ 1 + 2nπ for some integer n Z. The principal value Arg z of arg z is defined as the unique value θ arg z with π < θ π. So arg z = {Arg z + 2nπ : n Z}. Arg z = π z is a negative real number. CS457 Lecture 4/12: Polar Form and Euler s Formula 3/17

10 Recall... Polar coordinates Example: z = 1 + i in polar form is: CS457 Lecture 4/12: Polar Form and Euler s Formula 4/17

11 Recall... Polar coordinates To find θ in general: If z 0, then y x = r sin θ r cos θ = sin θ = tan θ. cos θ Hence arg z tan 1 y x where tan 1 u = {v R : tan v = u}, u R. CS457 Lecture 4/12: Polar Form and Euler s Formula 5/17

12 Recall... Polar coordinates Example: Express z = 2 2i in polar form. CS457 Lecture 4/12: Polar Form and Euler s Formula 6/17

13 Euler s formula. For θ R define Euler s 1 Formula: exp(i θ) = e i θ = cos θ + i sin θ So can write the polar form of z 0 as z = r (cos θ + i sin θ) = r e i θ. Exercise: Show that d dθ (e i θ ) = d dθ (cos θ + i sin θ) 1 Leonhard Euler, CS457 Lecture 4/12: Polar Form and Euler s Formula 7/17

14 Euler s formula. A little History Euler s formula is actually due to Roger Cotes in His form was log(cos x + i sin x) = ix. Euler published the equation in its current form in Neither of them used the geometrical interpretation of the formula that we have used. That was first given 50 years later by Caspar Wessel. Cotes is also famous for the Newton-Cotes methods for estimating definite integrals, the most famous examples of which are the Trapezium Rule and Simpson s Rule. CS457 Lecture 4/12: Polar Form and Euler s Formula 8/17

15 Euler s formula. cos x =... We can use Euler s formula to show that cos(x) = 1 e i 2( x + e ix), as follows: Exer: Show that sin x = 1 2( e i x e ix). CS457 Lecture 4/12: Polar Form and Euler s Formula 9/17

16 Euler s formula. cos x =... We can use Euler s formula to show that cos(x) = 1 e i 2( x + e ix), as follows: Exer: Show that sin x = 1 2( e i x e ix). CS457 Lecture 4/12: Polar Form and Euler s Formula 9/17

17 Inverse, Products, and Quotients Suppose z = re i θ, z 1 = r 1 e i θ 1 and z 1 = 1 z = 1 re i θ = 1 r e i θ, z 1 z 2 = r 1 r 2 e i (θ 1+θ 2 ) z 1 z 2 = r 1 r 2 e i (θ 1 θ 2 ). Multiplication in polar form is...: and z 2 = r 2 e i θ 2. Then CS457 Lecture 4/12: Polar Form and Euler s Formula 10/17

18 Inverse, Products, and Quotients Suppose z = re i θ, z 1 = r 1 e i θ 1 and z 1 = 1 z = 1 re i θ = 1 r e i θ, z 1 z 2 = r 1 r 2 e i (θ 1+θ 2 ) z 1 z 2 = r 1 r 2 e i (θ 1 θ 2 ). Multiplication in polar form is...: and z 2 = r 2 e i θ 2. Then CS457 Lecture 4/12: Polar Form and Euler s Formula 10/17

19 Binomial Coefficients. The Binomial Theorem: ( ) ( ) ( ) n n n (a+b) n = a n + a n 1 b+ a n 2 b ab n 1 +b n 1 2 n 1 More compactly: (a + b) n = n k=0 ( ) n a n k b k, k Here the Binomial Coefficient ( n k) ( n choose k ) is ( ) n n! = k k! (n k)!, and n! ( n factorial ) is n! = n (n 1) CS457 Lecture 4/12: Polar Form and Euler s Formula 11/17

20 Binomial Coefficients. The Binomial Theorem: ( ) ( ) ( ) n n n (a+b) n = a n + a n 1 b+ a n 2 b ab n 1 +b n 1 2 n 1 More compactly: (a + b) n = n k=0 ( ) n a n k b k, k Here the Binomial Coefficient ( n k) ( n choose k ) is ( ) n n! = k k! (n k)!, and n! ( n factorial ) is n! = n (n 1) CS457 Lecture 4/12: Polar Form and Euler s Formula 11/17

21 Some combinatorics n! Permutations: There are (n k)! ways of picking and ordered collection of k objects from n Combinations: There are ( ) n k = n! k!(n k)! ways of choosing, without order, k objects from n ( ) ( ) ( ) n n 1 n 1 Pascal s Triangle: = +. k k 1 k n ( ) n Sum (a = b = 1): = 2 n. k k=0 CS457 Lecture 4/12: Polar Form and Euler s Formula 12/17

22 Some combinatorics n! Permutations: There are (n k)! ways of picking and ordered collection of k objects from n Combinations: There are ( ) n k = n! k!(n k)! ways of choosing, without order, k objects from n ( ) ( ) ( ) n n 1 n 1 Pascal s Triangle: = +. k k 1 k n ( ) n Sum (a = b = 1): = 2 n. k k=0 CS457 Lecture 4/12: Polar Form and Euler s Formula 12/17

23 Some combinatorics n! Permutations: There are (n k)! ways of picking and ordered collection of k objects from n Combinations: There are ( ) n k = n! k!(n k)! ways of choosing, without order, k objects from n ( ) ( ) ( ) n n 1 n 1 Pascal s Triangle: = +. k k 1 k n ( ) n Sum (a = b = 1): = 2 n. k k=0 CS457 Lecture 4/12: Polar Form and Euler s Formula 12/17

24 Some combinatorics n! Permutations: There are (n k)! ways of picking and ordered collection of k objects from n Combinations: There are ( ) n k = n! k!(n k)! ways of choosing, without order, k objects from n ( ) ( ) ( ) n n 1 n 1 Pascal s Triangle: = +. k k 1 k n ( ) n Sum (a = b = 1): = 2 n. k k=0 CS457 Lecture 4/12: Polar Form and Euler s Formula 12/17

25 Some combinatorics De Moivre s Theorem 2 is: (cos θ + i sin θ) n = cos nθ + i sin nθ for all n Z. Proof (1): Use Euler s formula: 2 Abraham de Moivre, CS457 Lecture 4/12: Polar Form and Euler s Formula 13/17

26 Some combinatorics De Moivre s Theorem 2 is: (cos θ + i sin θ) n = cos nθ + i sin nθ for all n Z. Proof (1): Use Euler s formula: 2 Abraham de Moivre, CS457 Lecture 4/12: Polar Form and Euler s Formula 13/17

27 Some combinatorics Proof (2): Exercise Use an inductive proof and the fact that cos α cos β sin α sin β = cos(α + β) and sin α cos β + cos α sin β = sin(α + β) CS457 Lecture 4/12: Polar Form and Euler s Formula 14/17

28 Applications of de Moirve s Theorem: De Moivre s Theorem can be used to express cos or sin of multiple angles in terms of powers of cos θ and sin θ. Example: Show that cos 3θ = 4 cos 3 θ 3 cos θ. (Note: cos 2 θ is short-hand for (cos θ) 2 ) (Hint: Also need to use that cos 2 θ = 1 sin 2 θ) CS457 Lecture 4/12: Polar Form and Euler s Formula 15/17

29 Applications of de Moirve s Theorem: The theorem can be used to express powers of cos θ and sin θ in terms of cos or sin of multiple angles. Example: Show that cos 2 θ = 1 2 (cos 2θ + 1). CS457 Lecture 4/12: Polar Form and Euler s Formula 16/17

30 Applications of de Moirve s Theorem: Exercise: Show that 16 cos 4 θ = 2 cos 4θ + 2 cos 2θ + 6. (Hint: use the previous identity for cos 2 θ. CS457 Lecture 4/12: Polar Form and Euler s Formula 17/17