On expa+ib

Euler's identity, eiθ = cos(θ) + i sin(θ) is useful in various ways. It is discussed with a circular plot of it on the Argand plane, which you can see at my expiθ web page.

In addition, given that expa+b = expa*expb, we often use expa+ib. Some uses of this are shown here, including a plot of it as a circle on the Argand plane.

Often we encounter functions involving eatsin(bt) or eatcos(bt), and want to differentiate or (and this is harder) integrate them. Applications include Fourier series, transforms, Laplace transforms, etc.

We show here that using Euler's identity can make this easier - and in fact one can often process both functions in one go. Complex numbers make life easier!

If want, say, the differential of eat sin(bt), the concept is to find the differential of e(a+ib)t which gives the differential of both eat cos(bt) and eat sin(bt), the real part being that of eat cos(bt) the imaginary part that for eat sin(bt). One then equates the real and imaginary parts, and so finds two differentials at once.

This concept is even more powerful for integration, as is known if you have tried integration by parts!

e(a + ib)θ on the Argand plane.

This involves sinusoids whose amplitude varies : typically a is negative, so the amplitude decays. You can change a and b and see the result. The actual plot is exp(aθ) exp(i 2 π bθ) to correspond with rotations being multiples of 2π radians.

a (-5..0) b (0..5)