Math144Notes

# Math144Notes - Advanced Engineering Math II Math 144...

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Advanced Engineering Math II Math 144 Lecture Notes by Stefan Waner (First printing: 2003) Department of Mathematics, Hofstra University

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2 1. Algebra and Geometry of Complex Numbers (based on §§17.1–17.3 of Zill) Definition 1.1 A complex number has the form z = (x, y), where x and y are real numbers. x is referred to as the real part of z, and y is referred to as the imaginary part of z . We write Re (z) = x , Im (z) = y. Denote the set of complex numbers by C I . Think of the set of real numbers as a subset of C I by writing the real number x as (x, 0) . The complex number (0, 1) is called i. Examples 3 = (3, 0), (0, 5), (-1, - π ), i = (0, 1). Geometric Representation of a Complex Number- in class. Definition 1.2 Addition and multiplication of complex numbers, and also multiplication by reals are given by: (x, y) + (x ' , y ' ) = ((x+x ' ), (y+y ' )) (x, y)(x ' , y ' ) = ((xx ' -yy ' ), (xy ' +x ' y)) (x, y) = ( x, y). Geometric Representation of Addition- in class. (Multiplication later) Examples 1.3 (a) 3+4 = (3, 0)+(4, 0) = (7, 0) = 7 (b) 3 4 = (3, 0)(4, 0) = (12-0, 0) = (12, 0) = 12 (c) (0, y) = y(0, 1) = yi (which we also write as iy). (d) In general, z = (x, y) = (x, 0) + (0, y) = x + iy. z = x + iy (e) Also, i 2 = (0, 1)(0, 1) = (-1, 0) = -1. i 2 = -1 (g) 4 - 3i = (4, -3) . Note In view of (d) above, from now on we shall write the complex number (x, y) as x+iy . Definitions 1.4 The complex conjugate , z , of the complex number z = x+iy given by z = x - iy. The magnitude , |z| of z = x+iy is given by |z| = x 2 +y 2 . Examples and Geometric Representation of Conjugation and Magnitude - in class. Notes 1. z + z = (x+iy) + (x-iy) = 2x = 2 Re (z) . Therefore, Re (z) = 1 2 (z+z)
3 z - z = (x+iy) - (x-iy) = 2iy = 2i Im (z) . Therefore, Im (z) = 1 2i (z-z) 2. Note that zz = (x+iy)(x-iy) = x 2 -i 2 y 2 = x 2 +y 2 = |z| 2 zz = |z| 2 3 . If z 0 , then z has a multiplicative inverse. Why? because: z · z |z| 2 = zz |z| 2 = |z| 2 |z| 2 = 1. Hence, z -1 = z |z| 2 Examples (a) 1 i = -i (b) 1 3+4i = 3-4i 25 (c) 1 1 2 (1+i) = 1 2 (1-i) (d) 1 cos ø + i sin ø = cos (-ø) + i sin (-ø) 4. There is also the Triangle Inequality: |z 1 + z 2 | |z 1 | + |z 2 |. Proof We square both sides and compare them. Write z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 . Then |z 1 + z 2 | 2 = (x 1 +x 2 ) 2 + (y 1 +y 2 ) 2 = x 1 2 + x 2 2 + 2x 1 x 2 + y 1 2 + y 2 2 + 2y 1 y 2 . On the other hand, (|z 1 | + |z 2 |) 2 = |z 1 | 2 + 2|z 1 ||z 2 | + |z 2 | 2 = x 1 2 + x 2 2 + y 1 2 + y 2 2 + 2|z 1 ||z 2 |. Subtracting, (|z 1 | + |z 2 |) 2 - |z 1 + z 2 | 2 = 2|z 1 ||z 2 | - 2(x 1 x 2 + y 1 y 2 ) = 2 [ |(x 1 ,y 1 )||(x 2 ,y 2 )| - (x 1 ,y 1 ).(x 2 ,y 2 ) ] (in vector form) = 2 [ |(x 1 ,y 1 )||(x 2 ,y 2 )| - |(x 1 ,y 1 )||(x 2 ,y 2 )| cos ] = 2 |(x 1 ,y 1 )||(x 2 ,y 2 )| (1 - cos ) 0, giving the result. Note The triangle inequality can also be seen by drawing a picture of z 1 + z 2 . 5. We now consider the polar form of these things: If z = x+iy, we can write x = r cos ø and y = r sin ø, getting z = r cos ø + ir sin ø, so This is called the polar form of z . It is important to draw pictures in order to feel comfortable with the polar representation. Here r is the magnitude of z, r = |z| , and ø is called the argument of z , denoted arg (z). To calculate ø , we can use the fact that tan ø = y/x. Thus ø is not arctan (y/x) as claimed in the book, but by: since the arctan function takes values between - π /2 and π /2. The principal value of arg (z) is the unique choice of ø such that - π < ø π . We write this as Arg (z) - π < Arg (z) ≤ π - π ≤ A

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4 Examples (a) Express z = 1+i in polar form, using the principal value (b) Same for 3 + 3 3 i (c) 6 = 6( cos 0 + i sin 0) 6.
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The provided text is a set of lecture notes on Advanced Engineering Math, specifically covering the algebra and geometry of complex numbers. It includes definitions, examples, and explanations of complex numbers, their operations, and their geometric representations.
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Math144Notes

# Math144Notes - Advanced Engineering Math II Math 144...

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Advanced Engineering Math II Math 144 Lecture Notes by Stefan Waner (First printing: 2003) Department of Mathematics, Hofstra University

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2 1. Algebra and Geometry of Complex Numbers (based on §§17.1–17.3 of Zill) Definition 1.1 A complex number has the form z = (x, y), where x and y are real numbers. x is referred to as the real part of z, and y is referred to as the imaginary part of z . We write Re (z) = x , Im (z) = y. Denote the set of complex numbers by C I . Think of the set of real numbers as a subset of C I by writing the real number x as (x, 0) . The complex number (0, 1) is called i. Examples 3 = (3, 0), (0, 5), (-1, - π ), i = (0, 1). Geometric Representation of a Complex Number- in class. Definition 1.2 Addition and multiplication of complex numbers, and also multiplication by reals are given by: (x, y) + (x ' , y ' ) = ((x+x ' ), (y+y ' )) (x, y)(x ' , y ' ) = ((xx ' -yy ' ), (xy ' +x ' y)) (x, y) = ( x, y). Geometric Representation of Addition- in class. (Multiplication later) Examples 1.3 (a) 3+4 = (3, 0)+(4, 0) = (7, 0) = 7 (b) 3 4 = (3, 0)(4, 0) = (12-0, 0) = (12, 0) = 12 (c) (0, y) = y(0, 1) = yi (which we also write as iy). (d) In general, z = (x, y) = (x, 0) + (0, y) = x + iy. z = x + iy (e) Also, i 2 = (0, 1)(0, 1) = (-1, 0) = -1. i 2 = -1 (g) 4 - 3i = (4, -3) . Note In view of (d) above, from now on we shall write the complex number (x, y) as x+iy . Definitions 1.4 The complex conjugate , z , of the complex number z = x+iy given by z = x - iy. The magnitude , |z| of z = x+iy is given by |z| = x 2 +y 2 . Examples and Geometric Representation of Conjugation and Magnitude - in class. Notes 1. z + z = (x+iy) + (x-iy) = 2x = 2 Re (z) . Therefore, Re (z) = 1 2 (z+z)
3 z - z = (x+iy) - (x-iy) = 2iy = 2i Im (z) . Therefore, Im (z) = 1 2i (z-z) 2. Note that zz = (x+iy)(x-iy) = x 2 -i 2 y 2 = x 2 +y 2 = |z| 2 zz = |z| 2 3 . If z 0 , then z has a multiplicative inverse. Why? because: z · z |z| 2 = zz |z| 2 = |z| 2 |z| 2 = 1. Hence, z -1 = z |z| 2 Examples (a) 1 i = -i (b) 1 3+4i = 3-4i 25 (c) 1 1 2 (1+i) = 1 2 (1-i) (d) 1 cos ø + i sin ø = cos (-ø) + i sin (-ø) 4. There is also the Triangle Inequality: |z 1 + z 2 | |z 1 | + |z 2 |. Proof We square both sides and compare them. Write z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 . Then |z 1 + z 2 | 2 = (x 1 +x 2 ) 2 + (y 1 +y 2 ) 2 = x 1 2 + x 2 2 + 2x 1 x 2 + y 1 2 + y 2 2 + 2y 1 y 2 . On the other hand, (|z 1 | + |z 2 |) 2 = |z 1 | 2 + 2|z 1 ||z 2 | + |z 2 | 2 = x 1 2 + x 2 2 + y 1 2 + y 2 2 + 2|z 1 ||z 2 |. Subtracting, (|z 1 | + |z 2 |) 2 - |z 1 + z 2 | 2 = 2|z 1 ||z 2 | - 2(x 1 x 2 + y 1 y 2 ) = 2 [ |(x 1 ,y 1 )||(x 2 ,y 2 )| - (x 1 ,y 1 ).(x 2 ,y 2 ) ] (in vector form) = 2 [ |(x 1 ,y 1 )||(x 2 ,y 2 )| - |(x 1 ,y 1 )||(x 2 ,y 2 )| cos ] = 2 |(x 1 ,y 1 )||(x 2 ,y 2 )| (1 - cos ) 0, giving the result. Note The triangle inequality can also be seen by drawing a picture of z 1 + z 2 . 5. We now consider the polar form of these things: If z = x+iy, we can write x = r cos ø and y = r sin ø, getting z = r cos ø + ir sin ø, so This is called the polar form of z . It is important to draw pictures in order to feel comfortable with the polar representation. Here r is the magnitude of z, r = |z| , and ø is called the argument of z , denoted arg (z). To calculate ø , we can use the fact that tan ø = y/x. Thus ø is not arctan (y/x) as claimed in the book, but by: since the arctan function takes values between - π /2 and π /2. The principal value of arg (z) is the unique choice of ø such that - π < ø π . We write this as Arg (z) - π < Arg (z) ≤ π - π ≤ A

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4 Examples (a) Express z = 1+i in polar form, using the principal value (b) Same for 3 + 3 3 i (c) 6 = 6( cos 0 + i sin 0) 6.
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The provided text is a set of lecture notes on Advanced Engineering Math, specifically covering the algebra and geometry of complex numbers. It includes definitions, examples, and explanations of complex numbers, their operations, and their geometric representations.

TYPE: NO_QUESTION

The text does not contain any specific mathematical questions or problems for the reader to solve. It is primarily expository, providing definitions, examples, and explanations.