A Level Maths Session
Inverse Function  Exponential Function  Maclaurin Series  Complex Number Exponential Form 
Complex Number Exponential Form
Hi Everyone,
Welcome to this session.
proof complex number exponential form
Exercises
Please try these questions.
Given 2+3i. Treat 2 (the real part) as an x coordinate and 3 (the imaginary part) as a y coordinate.
 Sketch the x,y axes and mark this point on the graph. Label it P.
 Draw the line OP and find its length r. (Call r the “modulus”).
 Find the angle θ between OP and the x axis. (Call θ the “argument”).
 Rewrite 2+3i in the form r(cosθ + i sinθ). (Called the polar form).

Prove that cosθ + i sinθ = e^{iθ} using these steps:
 Write out the first 4 terms in the Maclaurin’s series for cosθ, sinθ and e^{iθ}.
 Show for the first 4 terms that cosθ + i sinθ = e^{iθ}. (Called Euler’s formula.)
 Given 2+3i. Write this in the form r e^{iθ}. Call this the exponential form.
Past Exam Questions
In this video, I shall go through some past exam questions.
Thanks for attending this session.
Dr Hock
9 May 2014