Complex Numbers in Engineering Mathematics: Geometric Interpretation and Properties, Lecture notes of Engineering Science and Technology

Download Complex Numbers in Engineering Mathematics: Geometric Interpretation and Properties and more Lecture notes Engineering Science and Technology in PDF only on Docsity!

MAT 247 Engineering Mathematics

Hakkı Ula¸s Unal¨ Dept. of Electrical-Electronics Eng. Eski¸sehir Technical University, Turkey

October 24, 2018

Today

Complex Numbers

Analytic functions

Derivatives

Geometric Interpretation of Complex numbers

Since summation of two complex numbers z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 equals z 1 + z 2 = (x 1 + x 2 ) + i(y 1 + y 2 ), then, its geometric representation can be considered as sum of two vectors.

z 1

z 2 x

y ◦

z 1 + z 2

Geometric Interpretation of Complex numbers

The modulus or absolute value of a complex number z = x + yi is defined as |z| =

x^2 + y^2 , which corresponds to distance btw the point z and origin.

Geometric Interpretation of Complex numbers

The modulus or absolute value of a complex number z = x + yi is defined as |z| =

x^2 + y^2 , which corresponds to distance btw the point z and origin.

IT IS OBVIOUS THAT z 1 < z 2 IS MEANINGLESS,

however, |z 1 | < |z 2 |, is meaningful, since it compares the distance of each complex point to the origin. The distance btw two complex numbers z 1 and z 2 is

|z 1 − z 2 |

Geometric Interpretation of Complex numbers

By definition of the modulus (or absolute value) of a complex number z = x + yi, since

|z|^2 = x^2 + y^2 = Re(z)^2 + Im(z)^2 ,

implies that |z| ≥ Re(z) |z| ≥ Im(z)

Note that z = z |z| = |z|,

Note that z = z |z| = |z|,

Let z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2. Then, since

z 1 + z 2 = (x 1 + x 2 ) − i(y 1 + y 2 ) = (x 1 − iy 1 ) + (x 2 − iy 2 ) = z 1 + z 2

Note that z = z |z| = |z|,

Let z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2. Then, since

z 1 + z 2 = (x 1 + x 2 ) − i(y 1 + y 2 ) = (x 1 − iy 1 ) + (x 2 − iy 2 ) = z 1 + z 2

Similarly, since

z 1 z 2 = (x 1 x 2 − y 1 y 2 ) − i(x 1 y 2 + x 2 y 1 ) = (x 1 − iy 1 )(x 2 − iy 2 ),

z 1 z 2 = z 1 z 2

( z 1 z 2

z 1 z 2

Let z = x + jy. Then,

z + z =? z − z =?,

Let z = x + jy. Then,

z + z =? z − z =?,

Re(z) =

z + z 2 Im(z) =

z − z 2 i

Let z = x + jy. Then, zz =?,

Let z = x + jy. Then,

z + z =? z − z =?,

Re(z) =

z + z 2 Im(z) =

z − z 2 i

Let z = x + jy. Then, zz =?,

zz = |z|^2

Some Remarks on Moduli

Triangle Ineq

Let z 1 and z 2 be two complex numbers, then,

|z 1 + z 2 | ≤ |z 1 | + |z 2 |,

Let z 1 and z 2 be two complex numbers, then, by Triangle Ineq.,

||z 1 | − |z 2 || ≤ |z 1 + z 2 | ≤ |z 1 | + |z 2 |,