Complex numbers and Euler's formula
First: review "radians"
The definition of the radian is:
The measure of a central angle, $\theta$
in radians is the ratio of the length of the circular arc $s$
to the radius $r$
of the circle.
$$\theta = \frac sr.$$
Therefore, on the unit circle, which has a radius of $r=1$, the angle $\theta$ is equal to the arclength $s$.
Please stare at this excellent Wikimedia image for a few cycles of the animation,
based on the definition of 1 radian:
An arc of a circle with the same length as the radius of that circle subtends an angle of 1 radian.
The conversion factor to memorize is: $$\frac 12\text{rotation through a circle}=\pi\text{ radians}=180{}^o$$
The Complex plane & Euler's Formula
Read over (This includes you, electronics students!) this brief and historical justification of Euler's formula.
$$e^{i\phi}=\cos\phi+i\sin\phi.$$
This holds when the angle $\phi$ is in radians.
This material on Taylor series (mathphys) shows how to get the series approximation of $\sin x$ by Taylor-expanding the sine function about zero: $$\sin(x)=x -x^3/3!+x^5/5!-x^7/7!+x^9/9!-....$$ and visualizing how the series approaches $\sin x$.
Sine waves, using the Euler function
It turns out to be tremendulously useful to represent a time-varying sine-wave as the real part of a complex Euler function, for example, consider the complex function $z(t) = 7e^{i5t}$. To find the real part, $\text{Re}$[...] $$\begineq V(t)&=\text{Re}\left[7e^{i5t}\right]\\ &= \text{Re} \left[7\left(\cos(5t)+i\sin(5t)\right)\right]\\ &= \text{Re} \left[7\cos(5t)+i7\sin(5t)\right]\\ &=\color{blue}7\cos(5t). \endeq $$ [BTW, don't worry that this is a cosine function instead of a sine function: The shape of a cosine function is the same as the shape of a sine function, just shifted over a bit. Both functions describe generic "sine-waves".]
- the amplitude
- the period, $\tau$, assuming that $t$ is measured in seconds.
- the frequency, $f$, in Hz (cycles/sec).
Now Check your answers: The argument of the cosine function, $5t$, should be in radians. To make this come out right, since $t$ is in seconds, the number 5 must have units of radians/second. There are are $2\pi$ radians in one "cycle" of a circle. (one repetition of the cosine function).
Convert 5 radians/second to some number of cycles/second using the conversion factor $2\pi$ radians = 1 cycle. Is this close to your estimated frequency in cycles/second? If so, what is a general formula for a cosine function $\cos(\omega t)$ connecting $\omega$ and $f$ (frequency)?
The complex plane
Perhaps you'll recall that you can represent a complex number as a vector in 2 dimensional space where the real portion is the $x$-component and the imaginary part is the $y$-component of the vector. Here we have a depiction of the complex number $-2 + 4i$: The blue dot has Cartesian components $(x,y)=(-2,4)$.
The position vector with components $(-2,4)$ is also shown on there as the blue arrow.
We can talk about a vector in terms of its length and direction. This is the way that a vector is described in polar coordinates: This position vector:
- has a length of $r=\sqrt{x^2+y^2}=\sqrt{(-2)^2+(4)^2}=\sqrt{20}\approx 4.47$,
- makes an angle (taking into account which quadrant it's in) of $\phi=\pi - \arctan(y/x)=\pi-\arctan(-4/2)=$ 2.03 radians ($\approx$ 117${}^o$) with the positive $x$ axis.
Euler's formula is $e^{i \phi } = \cos \phi + i \sin \phi$. More generally, we can write this as $re^{i \phi } = r\cos \phi + i r\sin \phi$ where the real and imaginary components of a complex number are connected to $r$ and $\phi$ in the formula like the $x$ and $y$ coordinates are connected through a conversion to polar coordinates $r$ and $\phi$.
So, apparently $-2+4i = 4.47e^{i 2.035}$ are equivalent ways of writing the same complex number.
1.) Sketch the complex number $\xi=3-4i$ in the complex plane. ['$\xi$' is the greek letter 'xi' pronounced xsi as in "taxi cab".] Then using cartesian$\to$polar conversions, write $\xi$ in the form $r\,e^{i\phi}$, where $r$ and $\phi$ themselves are real numbers.
$$r=\sqrt{3^2+(-4)^2}=5;\ \ \phi=arctan(-4/3)=-0.927$$ and therefore: $$3-4i=5e^{-i 0.927}.$$ Your calculator will tell you that -0.927 radians is -53${}^o$. And your sketch of the point (3, -4) should tell you that the vector is in the fourth quadrant.
2.) Sketch the complex number $\xi=5e^{i\frac{13\pi}{4}}$ in the complex plane. Then using polar$\to$cartesian conversions, write $\xi$ in the form $X+Yi$, where $X$ and $Y$ are real numbers.
A vector of length 5. The (counterclockwise) angle of rotation, $\phi=\frac{13}{4}\pi=(1+\frac 58)2\pi$ indicates that we must make 1 full rotation, and then 5/8 more of a rotation from the $x$-axis, so the vector is in quadrant III, pointing 45 degrees below the $-x$ direction. $$X=5*cos[13\pi/4]=-3.54; \ \ Y=5*sin[13\pi/4]=-3.54$$ so $$5e^{i\frac{13\pi}{4}}=-3.54 -3.54i.$$ From the right triangle, you can perhap also see that, since $X^2=Y^2$, we'd have $2X^2=25\Rightarrow |X|=\sqrt{25/2}=3.54$.
3.) Consider the function $\xi(t)=e^{i4t}$. In the complex plan, this describes a position vector of a constant length, 1. And its tip traces out a circle as time progresses.
Draw the trajectory of the tip of the vector in the complex plane. Indicate where on the circle the tip of the vector is at $t=0$. Figure out how much time it takes for the tip of the circle to return to its starting point. (This is the period, "$\tau$".) Also figure out whether the tip is rotating clockwise or counterclockwise around the origin.
$\xi$ is equal to 1 times the imaginary exponential. So the tip of the vector describes a circle of radius one.
At $t=0$, $\xi(0)=e^{i0}=1$. So, the vector is pointing at $X+iY=1+i0$, 1 on the real ($x$) axis.
When the angle in $e^i\phi$ increases to $2\pi$, one rotation is complete, and the vector is pointing in the $x-direction again. This will happen after a time $T$ fulfilling:
$$4T=2\pi\Rightarrow T=\pi/2.$$
The tip rotates counterclockwise. You can graph out a couple angles to see...
4.) Consider the function $\xi(t)=4e^{i\pi/6}e^{-i2t}$. In the complex plan, sketch out the circular trajectory that the tip of the position vector $\xi(t)$ traces out with time. What is its radius? Indicate where on the circle the tip of the vector is at $t=0$. Figure out how much time it takes for the tip of the circle to return to its starting point. (This is the period, "$T$".) Also figure out whether the tip is rotating clockwise or counterclockwise.
A position vector in the complex plane traces out a circle of radius 4. It starts pointing 30${}^0$ ($\pi/6$ radians) above the $x$ axis, and moves clockwise. When $2T=2\pi\Rightarrow T=\pi$ seconds have passed, the vector will have completed one whole rotation since $t=0$.
5.) Using the same $\xi(t)$ defined in the last problem: Carry out the multiplication of the two exponentials, so that you can express the the function in terms of just one $e$ raised to a more complicated-looking power. Now use Euler's formula to write $\xi(t)$ as $X(t)+iY(t)$, where $X(t)$ is the real portion of this function and $Y(t)$, the imaginary portion of this function.
$$\xi(t)=4e^{-i2t+i\pi/6}=4e^{-i(2t-\pi/6)}=4\cos(-(2t-\pi/6))+4i\sin(-(2t-\pi/6)) .$$