Euler's Formula & Complex Plane

e^(iθ) = (cosθ + isinθ) Where: θ=wt e^(iwt) = [cos(wt) + isin(wt)] Euler's formula traces a circle in the complex number plane as the value of t increases over time . So it applies to oscillating motion which has an angular frequency(w) that traces a circle, as well as rotating and circular motion. The total expression is the exponential function which has a real component on the real horizontal axis, and an imaginary component on the imaginary(i) vertical axis. The Taylor series for this exponential function reveals the fact that the powers of i, are the reason this exponential function traces a perfect circle in the complex number plane. Where: [i⁰=1], [i¹=i], [i²=-1], [i³=-i] The negative coefficients keep the function from experiencing exponential growth; and the coefficient sequence [1, i, -1, -i] repeats every four terms and corresponds to a 90° vector rotation in the complex number plane for each term. Which completes a full circle of 360°. The fact that an exponential funct...

Eulers Formula: e^(iθ) = (cosθ + isinθ) In the complex number plane, the real horizontal axis is labeled +real for the positive axis pointing to the right, and -real for the negative axis pointing to the right . The imaginary vertical axis is labeled +imaginary for the positive axis pointing up, and -imaginary for the negative axis pointing down. The values for theta(θ) range from 0 to 2π. So for each value of theta(θ) the corresponding point in the complex number plane has the coordinates: (cosθ,isinθ). Where cosθ corresponds to the real axis, and isinθ corresponds to the imaginary axis. When you plug in values for theta(θ) from 0 to 2π and plot the coordinates in the complex number plane you get a graph of a unit circle with the radius equal to one on the real axis. For: θ = 0°/ e^(iθ) = (1,0) = 1 For: θ = 90°/ e^(iθ) = (0,i) = i For: θ = 180°/ e^(iθ) = (-1,0) = -1 For: θ = 270°/ e^(iθ) = (0,-i) = -i You can see from these four points that the function traces a circle in the complex...

 For: θ = wt         w = 2π/T Where: T is the period for circular motion, which is the time it takes to complete one full cycle of motion.  So we can express the exponential function as a function of time(t) with a radius R on the real axis: F(t)ᵣₒₜ = Re^(iwt) As the value of t increases, this function traces a circle of radius R in the complex number plane. Which describes a rotating vector of magnitude R, that points away from the center, and rotates counterclockwise as t increases. On the real axis the function equals R which has units of length. The exponential function[e^(iwt)] is dimensionless so the function has units of length. When we take the derivative of this function with respect to time we get units for speed. So the derivative of the function gives us a velocity vector which represents circular motion. F(t)ᵣₒₜ = Re^(iwt) F'(t)ᵥₑₗ = iwRe^(iwt) Now this new function has units of R/sec which is dist/sec or speed. We notice this new exponential f...

When we take the second derivative with respect to time, we get a function having units of velocity divided by time or acceleration. But before we look at the acceleration function for circular motion, let's get straight the concepts of centrifugal acceleration and centripetal acceleration. This will allow us to better understand what the acceleration function is describing. When an object is traveling in a circle at a constant speed. A centripetal force is continuously acting on the object pulling it inward along the radius towards the center source, which counters the object's momentum to travel in a straight line. It's this tendency to travel straight that creates the effect of an outward push that the object experiences; but this effect is not the centrifugal force. The actual centrifugal force is equal and opposite the centripetal force, and exerts a pull on the center source, which comes from the object in motion trying to travel in a straight line. So the centripetal...

Now let's look at the acceleration function for circular motion, which is the second derivative of the rotational function with respect to time: F(t)ᵣₒₜ = Re^(iwt) F'(t)ᵥₑₗ = iwRe^(iwt) F''(t)ₐ = -w²Re^(iwt) The units here are R/w² which is dist/sec² for acceleration. We notice that the function is being multiplied by the -1 operator. Which means the acceleration vector is opposite the original rotational vector on the real axis. When we take the tail end of the acceleration vector and attach it to the front tip of the rotational vector; we notice that the acceleration points inward towards center as the rotational vector traces a circle. This then is the centripetal acceleration which acts on an object in circular motion, keeping it from traveling in a straight line tangent to the circle. Because of Newton's third law of reactionary forces, the centripetal acceleration causes an equal and opposite acceleration called the centrifugal acceleration which is a real for...

The imaginary unit i=√-1, is not a real number, it is a math construct that intertwines with real numbers in the complex number plane. This intertwining produces real measurable effects in the physical world without in fact being real. The first derivative of the exponential function in Euler's formula reveals that circular motion is encoded with the imaginary unit i, here it represents a direction perpendicular to the circle's radius at every point on the circle. So linear motion is actually the limiting case of circular motion. Where you get piecewise linearity for very small sections of the circle(sect→0). And linearity over larger sections as the radius increases(r→∞). This is why the kinetic energy formula for linear motion applies to circular motion as well: Krot = ½Iw² Where: w is angular velocity and I is moment of inertia  A point mass m at radius r has a moment of inertia(I): I = mr² and v = wr Krot = ½mr²w² = ½mv² By keeping the encoded imaginary unit i, we get: K...

Below is a list of some measurable effects that emerge from circular motion, but in fact are not physically real: 1) Reactance in ac circuits from oscillating periodic motion. It impedes the flow of ac current, but is not a physical property of the circuit element producing it like resistance is. 2) Time dilation from the circular flow of time; by making the speed of light invariant, you are saying that the total flow of motion through spacetime is c, whether it's motion in the time dimension, the spatial dimensions, or both; this total flow of motion is the circular flow of time following spacetime curvature. 3) The effect in lump ac circuits that allows you to view the sinusoidal source as instantaneously traveling through the circuit—even though the wavelength is much longer than the closed loop circuit path it travels. 4) The outward push that an object in circular motion experiences which is mistakenly taken as the centrifugal force; this is an effect from the momentum of the ...