## Complex Numbers and Ordinary Differential Equations

James Binney,
Michaelmas Term 2002
The course introduces complex numbers and differential equations at a level
that will prepare student for the Maths 2 paper in Natural Sciences Prelims.

Three problem sets will be issued

### 1 Complex Numbers

- 1.1 Why complex nos?
- 1.2 Argand diagram (complex plane)
- 1.3 Simple functions of z and de Moivre's Theorem
- 1.3.1 Trigonometric identities
- 1.3.2 Graphical representation of multiplication & division
- 1.3.3 Summing series with de Moivre

- 1.4 Curves in the complex plane
- 1.5 Roots of polynomials
- 1.5.1 Special polyniomials
- 1.5.2 Characterizing a polynomial by its roots

### 2 Linear Differential Equations

- 2.1 Differential operators
- 2.1.1 Order of a differential operator
- 2.1.2 Linear operators
- 2.1.3 Arbitrary constants & general solutions

- 2.2 Inhomogeneous terms
- 2.3 First-order linear equations
- 2.4 Second-order linear equations
- 2.5 Equations with constant coeÆcients
- 2.5.1 Factorization of operators & repeated roots
- 2.5.2 Equations of higher order

- 2.6 Particular integrals
- 2.6.1 Polynomial h
- 2.6.2 Exponential f
- 2.6.3 Sinusoidal h
- 2.6.4 Exponentially decaying sinusoidal h Since we are handling sinusoids

### 3 Application to Oscillators

- 3.1 Transients
- 3.2 Steady-state solutions
- 3.2.1 Power input
- 3.2.2 Energy dissipated
- 3.2.3 Quality factor

### 4 Systems of Linear DE's with Constant Coefficients

- 4.1 LCR circuits
- 4.1.1 Time evolution of the LCR circuits

### 5 Non-Linear Equations

- 5.1 Homogenoeous equations
- 5.2 Exact equations
- 5.3 Equations solved by interchange of variables
- 5.4 Equations solved by linear transformation

### 6 Green's Functions

- 6.1 The Dirac Delta-function
- 6.2 Deffining the Green's function
- 6.3 Finding G