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1 MA 1165 - Lecture 26 4/22/09 1 Exponential Form As we have seen, if we have two complex numbers in polar form z1 = r1 ( cos(θ1 ) + i sin(θ1 ) ) (1) z2 = r2 ( cos(θ2 ) + i sin(θ2 ) ) , (2) z1 · z2 = r1 · r2 ( cos(θ1 + θ2 ) + i sin(θ1 + θ2 ) ) . (3) and and we multiply them together, we get One thing you may notice is that a multiplication has become, at least in part, an addition. We’ve seen this before with the exponential functions. In particular, ea · eb = ea+b . (4) So what should ea+bi (5) equal? Well, it doesn’t have to equal anything in particular, but if we want to extend the exponential function to the complex numbers, and we want this function to retain its basic properties, then it pretty much has to be defined so that ea+bi = ea · ebi = ea (cos(b) + i sin(b)). (6) In particular, eiθ = cos(θ) + i sin(θ). When we do this, our polar form for complex numbers becomes z = reiθ . (7) (8) This gives us a more compact polar form, but it also brings the exponential functions and the trig functions together. Any sort of “unification” is an indication that we’ve done something right. It also makes DeMoivre’s theorem easy to remember. Keep in mind that this is a purely mathematical definition. 2 Quiz 26A Write the following complex numbers in the new (exponential) polar form. 1. i. 2. −3. 3. 1 + i. 4. 3 − 2i. Write the following complex form in the regular (rectangular) form. 5. 3eπi . 6. 2eπi/2 . 7. eπ/3 . MA 1165 - Lecture 26 3 No Homework 26 2