ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday,...

9
ME 565, , Due on Friday, January 18, 2019 Homework #1 Exercise 1-1 Express the complex numbers in the form a + bi and Re : (a) 1 4-3i 1 4 - 3i = 1 4 - 3i 4+3i 4+3i = 4+3i 16 - 12i + 12 + 9 = 4+3i 25 = 4 25 + 3 25 i a = 4 25 ,b = 3 25 R = a 2 + b 2 =0.2 θ = arctan( 3 25 4 25 ) = arctan( 3 4 ) 0.64 1 4 - 3i = 4 25 + 3 25 i =0.2e i0.64 (b) 3 2 - 1 2 i 4 3 2 - 1 2 i ! 4 = z 4 = ( Re ) 4 R = s ( 3 2 ) 2 +( 1 2 ) 2 =1 θ = arctan( 3 2 - 1 2 )= - π 3 z 4 = e -i 4π 3 = cos(- 4π 3 )+ i sin( 4π 3 ) 3 2 - 1 2 i ! 4 = -0.5 - i 3 2 = e -i 4π 3 (c)i n i 2 ,i 3 ,i 4 ,i 5 , ... = z n z = i = Re R =1 θ = π 2 ( e i π 2 ) n = e i πn 2 i n = cos( πn 2 )+ i sin( πn 2 )= e i πn 2 Page 1 of 8

Transcript of ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday,...

Page 1: ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday, January 18, 2019 Homework #1 Exercise 1-1 Express the complex numbers in the form

ME 565, , Due on Friday, January 18, 2019 Homework #1

Exercise 1-1 Express the complex numbers in the form a+ bi and Reiθ:(a) 1

4−3i

1

4− 3i=

1

4− 3i

4 + 3i

4 + 3i=

4 + 3i

16− 12i+ 12 + 9=

4 + 3i

25=

4

25+

3

25i

a =4

25, b =

3

25

R =√a2 + b2 = 0.2

θ = arctan(325425

) = arctan(3

4) ≈ 0.64

1

4− 3i=

4

25+

3

25i = 0.2ei0.64

(b)(√

32− 1

2i)4

(√3

2− 1

2i

)4

= z4 =(Reiθ

)4

R =

√(

√3

2)2 + (

1

2)2 = 1

θ = arctan(

√3

2

−12

) = −π3

z4 = e−i4π3 = cos(−4π

3) + i sin(

3)(√

3

2− 1

2i

)4

= −0.5− i√

3

2= e−i

4π3

(c)in

i2, i3, i4, i5, ... = zn

z = i = Reiθ

R = 1

θ =π

2(eiπ2

)n= ei

πn2

in = cos(πn

2) + i sin(

πn

2) = ei

πn2

Page 1 of 8

Page 2: ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday, January 18, 2019 Homework #1 Exercise 1-1 Express the complex numbers in the form

ME 565, , Due on Friday, January 18, 2019 Homework #1

Exercise 1-2 Find all solutions of:(a)ez = i

ez = ea+ib

= eaeib

i = Reiθ

= eiπ2

eaeib = eiπ2

eib = eiπ2

z = i(π

2+ 2πn

), with n ∈ Z

(b)ez = −1

−1 = Reiθ

= eiπ

ez = eiπ

eaeib = eiπ

z = i(π + 2πn

), with n ∈ Z

Page 2 of 8

Page 3: ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday, January 18, 2019 Homework #1 Exercise 1-1 Express the complex numbers in the form

ME 565, , Due on Friday, January 18, 2019 Homework #1

Exercise 1-3 Find all solutions of:(a)z4 = 1

z4 =(Reiθ

)4

= R4ei4θ

1 = Reiθ

= ei(2πn)

R4ei4θ = ei(2πn), with n ∈ ZR = 14θ = 2πn

θ =πn

2z = ei

πn2 , with n ∈ Z

(b)z2 = 4i

z2 =(Reiθ

)2

= R2ei2θ

4i = Reiθ

= 4ei(π2

+2πn), with n ∈ ZR2ei2θ = 4ei(

π2

+2πn)

R2 = 4R = 2

2θ =π

2+ 2πn

θ =π

4+ πn

z = 2ei(π4

+πn), with n ∈ Z(c)z2 = 1− i

z2 =(Reiθ

)2

= R2ei2θ

1− i = Reiθ

=√

2ei(−π4

+2πn), with n ∈ ZR2ei2θ =

√2ei(−

π4

+2πn)

R2 =√

2

R = 214

2θ = −π4

+ 2πn

θ = −π8

+ πn

z = 214 e(−

π8

+πn), with n ∈ Z

Page 3 of 8

Page 4: ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday, January 18, 2019 Homework #1 Exercise 1-1 Express the complex numbers in the form
Page 5: ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday, January 18, 2019 Homework #1 Exercise 1-1 Express the complex numbers in the form
Page 6: ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday, January 18, 2019 Homework #1 Exercise 1-1 Express the complex numbers in the form
Page 7: ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday, January 18, 2019 Homework #1 Exercise 1-1 Express the complex numbers in the form
Page 8: ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday, January 18, 2019 Homework #1 Exercise 1-1 Express the complex numbers in the form

DATE : 17-Jan-2019 12:39

Table of ContentsRevision : 1.00 .................................................................................................................... 1MATLAB Ver: 9.2.0.556344 (R2017a) ................................................................................... 1FILENAME : hw1_pr6.m ..................................................................................................... 1HW 1, Problem 6, ME 565 Winter 2019 ................................................................................. 1

Revision : 1.00

MATLAB Ver: 9.2.0.556344 (R2017a)

FILENAME : hw1_pr6.m

HW 1, Problem 6, ME 565 Winter 2019clear all; close all;

x = 0:0.05:2;y = 0:0.05:1;[X,Y] = meshgrid(x,y);

V1 = pi*sin(pi*X).*cos(pi*Y);V2 = -pi*cos(pi*X).*sin(pi*Y);

init_posx = linspace(0,2,10);init_pos = [init_posx',0.5*ones(10,1)];part_color = jet(10);tspan = [0,10];

func = @(t,z) [pi*sin(pi*z(1)).*cos(pi*z(2)); -pi*cos(pi*z(1)).*sin(pi*z(2))];

figure(1)hold onquiver(X,Y,V1,V2)for jj=1:10 [t,z] = ode45(func,tspan,init_pos(jj,:)); plot(z(:,1),z(:,2),'color',part_color(jj,:),'linewidth',1.5); scatter(z(1,1),z(1,2),'k');endylim([0 1])xlim([0 2])xlabel('x axis')ylabel('y axis')

1

Page 9: ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday, January 18, 2019 Homework #1 Exercise 1-1 Express the complex numbers in the form

DATE : 17-Jan-2019 12:39

Published with MATLAB® R2017a

2