Differential Equations - Packet 01

Complex numbers

Arithmetic

A complex number may be written in ‘Cartesian form’ with distinct real and imaginary components, with . For addition (as ) and multiplication by a real number (as ) one operates on and just as the components of a vector. Complex numbers are multiplied by complex numbers using the formula . Thus if and , then we have:

Complex arithmetic Mathlet

Exponents

Complex numbers can be used as exponents in virtue of Euler’s Formula:

Deriving Euler’s Formula using power series

Recall the power series formula for :

Plug in , and recall the power series formulas for and :

Other complex exponents can be calculated from Euler’s Formula by using exponent rules:

Euler Formula Mathlet

Polar format

A complex number can be written in ‘polar form’ as . This translates to ‘Cartesian form’ using the exponential rule:

These forms allow one to graph complex numbers in the plane using the polar coordinates or the corresponding Cartesian coordinates .

The radius is called the modulus and the angle is called the argument of the complex number . The modulus of is the number . It is sometimes written .

If a complex number is graphed in the coordinate plane, and this number is multiplied by another complex number , then the product is given by scaling by , and rotating by (counterclockwise).

Complex exponentials Mathlet

Conjugate

Complex numbers come in natural pairs. Given , there is a conjugate number . Conjugates correspond to reflection in the real axis.

It is easy to verify a relationship between conjugation and multiplication:

Given , we sometimes write and for the real part and imaginary part. These can be extracted using conjugation:

Conjugation preserves algebraic operations:

Division

Division can be taken by complex numbers. The quotient can be calculated using conjugate multiplication. Supposing , then:

Roots

Whole number roots of complex numbers are found using the exponential form:

In polar form, this means the root is taken on the modulus directly, and the angle is divided by the root number.

This procedure gives the analogue of . But of course, as well. For any given , there are exactly distinct complex numbers, , such that . These numbers are given by adding to the angle, for various :

Complex roots Mathlet