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