I’m a pseudo Math geek. I understand the derivation of Euler’s Identity, but I want to understand this:
Three-dimensional visualization of Euler’s formula
Can someone help me explain it so when I print it and post it in my office, I can explain it to other people other than just saying “It’s cool”
If you have exp(iz) = cos(z) + isin(z), then:
You have the z-axis in the picture, and the function exp(iz) is a function of z. The x and y axes represent the real and imaginary parts of exp(iz). In other words the real part is cos(z), and the imaginary part (the part multiplied by ‘i’) is sin(z).
As you change z, the vector in the complex plane (x,y) sweeps along a circle, which makes sense because you can describe a circle by (x,y) = (cos(z),sin(z)).
Try thinking of it this way: for a real function of x, you can graph it with an x-axis and a y-axis.
For a complex function, one possible way to graph it is with a z axis for the input and an entire complex plane as the output “axis” because your output is a complex number with a real and imaginary part. So imagine that as you vary z, the complex plane moves along the z axis, and the output is plotted as a point on that plane with its real and imaginary parts. The plane then traces out a 3D curve as it moves about on the z axis.
Urk, that is a rotten diagram.
Anyway. What the diagram (clearly drawn by an engineer, since it uses j instead of i) shows is the 3 dimensional representation of the the expression of
e^iz
with z varying from 0 to 4pi
The vertical plane is the complex plane, so real number are the horizontal axis, imaginary numbers on the vertical axis.
When z = 0 you can see that the value is 1, or rather (1,0) as a complex number. That is where the red line starts at the top right of the diagram. z increases in value along the line from top right to bottom left. The plane formed by the z axis and the x axis is labelled as the real plane, and both z and x are real numbers.
You can see that as z increases the value of the expression e^iz rises up off the x axis gaining a positive imaginary component at the same time that the real component falls. If you stood at the end of the z axis and looked straight along it to the complex (x,j) plane you would see the value of e^iz move in a circle of radius 1 on the complex plane. The rest of the diagram is delving deeper in to the nature of this movement.
The two gray lines intertwined with the red line are the real and imaginary values of the value of e^iz, the red line is the complex number formed from that pair. What you will notice is that these two gray lines are a sine and cosine function of z. They represent this part of the identity.
*e*^*i*z = cos(z) + *i *sin(z)
You can just about see how the red line can be projected onto either the horizontal plane, or the vertical plane that forms the imaginary axis along the z axis, would lead to the two gray lines.
What you see is that as z increases the locus is the function is a spiral that spins around the z axis once for each 2pi increase in the value of z. For a value of pi for z, the spiral has made it half way around the circle, and the value is (-1,0) or just plain -1.
Okay… Thanks… it does make sense now… so… is there any better graph representing this online?
Visual representations of the function e^(it) are are ubiquitous, having already been placed in millions of rooms across the globe, and, indeed, many even wear portable versions. The common term for them is “(analog) clocks”.