1. Introduction: What is it? Euler"s formula is this crazy formula that ties exponentials to sinusoids with imaginary numbers: Does that make sense? It absolutely didn"t to me as soon as I initially saw it. What does it really expect to raise a number to an imaginary power? I think our instinct once reasoning about exponents is to imagine multiplying the base by itself "exponent" variety of times. So, for circumstances, 34 would be 3*3*3*3. But this line of thinking leads to a dead end as soon as we have an imaginary exponent, so we can not usage that meaning of exponentiation. We"ll have to use an additional definition to prove that this is true. So I will go ahead and also prove it two different methods to convince you that it provides feeling, and also then I"ll present just how it actually renders life much easier (in spite of how starray it may seem). 2. Proving it via a differential equation I"m going to begin via a simple proof of this making use of some standard calculus. I"ll begin by working backwards, beginning via the result: Now distinguish f(x) one time: Now note the complying with differential equation: < fracd f(x)dx = if(x) > As such, < f(x) = e^ix > And we"re done. That was easy! 3. Proving it by means of Taylor Series expansion Respeak to from my tutorial on Taylor Series that the Maclaurin Series of ex is: < e^x = sum_n=0^infty fracx^nn! > Plug in i*theta for x: < e^i heta = 1 + (i heta) + frac(i heta)^22! + frac(i heta)^33! + frac(i heta)^44! + frac(i heta)^55! + frac(i heta)^66! + ... > Now simplify all of the terms with powers of i in them, noting that i2 = -1: < e^i heta = 1 + (i heta) + - frac heta^22! -i frac heta^33! + frac heta^44! + i frac heta^55! - frac heta^66! + ... > Now group every one of the even order terms together and also all of the odd order terms together, noting that only the odd order terms have an i variable left: < e^i heta = (1 - frac heta^22! + frac heta^44! - frac heta^66! + ...) + i( heta - frac heta^33! + frac heta^55! - ...) > You should now identify a pattern here; all of the also powers create the Maclaurin series for cos(x), and every one of the odd powers form the Maclaurin series for sin(x). As such, we have actually proved once aacquire that eix = cos(x) + isin(x). 4. Visualizing Euler"s Formula So hopecompletely currently you can buy Euler"s formula; if not at a concrete level, at least at some abstract level by noting that every little thing "works" mathematically if you accept that i is the square root of -1. Now it"s tough to attempt to visualize it unless we come up via some conventions. For this, we first should remember exactly how to plot complicated numbers in rectangular form. We commonly visualize facility numbers in 2D by convention, plotting the genuine component on the horizontal axis, and also the imaginary part on the vertical axis. For instance, the complicated number, 4 + i, would certainly be plotted as follows: If we adapt this convention, we alert that the components of e^(ix) in the complex airplane are (cos(x), sin(x)). Hence, feeding various x values to Euler"s formula traces out a unit circle in the facility plane. In this manner, Euler"s formula deserve to be supplied to express facility numbers in polar form. We sindicate offer a magnitude, A, and an angle, theta, that a complex number makes through the real axis (the arc tangent of the imaginary over the genuine component), and also we have the right to express it utilizing Euler"s formula. For circumstances, we deserve to expush the number (1 + i) as: ...where A is sqrt(2) and theta is 45 degrees in this situation 5. Trig Identities and Euler"s Formula Euler"s Formula renders it really simple to prove complicated trig identities, bereason we currently get to use properties of exponents to aid us out. Let me present you a quick example: Proof: <eginalign e^i(u + v) &= cos(u + v) + isin(u + v) \ &= e^iue^iv \ &= \ &= cos(u)cos(v) + icos(u)sin(v) + isin(u)cos(v) - sin(u)sin(v) \ &= cos(u)cos(v) - sin(u)sin(v) + i(cos(u)sin(v) + sin(u)cos(v)) endalign> Because of this, we end up with: Now grouping both components together: < cos(u + v) = cos(u)cos(v) - sin(u)sin(v)> and also... To gain the other part of the "plus or minus," plug in (-v), noting that cos(-u) = cos(u) and sin(-u) = -sin(u) : and also... > The normal method to prove it via trig is a lot even more complicated (I don"t even feel it"s worth composing down below right now). But this identity is incredibly important, and also we will need it later on once we talk about phasors in one more tutorial. 6. Expressing Sine and Cosine One more quick note about exactly how to write sine and cosine in regards to euler"s identification.
These formulas depend on the truth that cosine is also (cos(x) = cos(-x)) and also sine is odd (sin(x) = - sin(-x)). The goal will be to usage these facts to our benefit to cancel out the sine as soon as we"re trying to gain the formula for the cosine, or vice versa: < sin(x) = frace^ix - e^-ix2i > Proof: <eginalign frace^ix - e^-ix2i &= fraccos(x) + isin(x) - (cos(-x) + isin(-x))2i\ &= fraccos(x) + isin(x) - (cos(x) - isin(x))2i \ &= fraccos(x) + isin(x) - cos(x) + isin(x)2i \ &= sin(x) endalign> < cos(x) = frace^ix + e^-ix2 > Proof: <eginalign frace^ix + e^-ix2 &= fraccos(x) + isin(x) + cos(-x) + isin(-x)2 \ &= fraccos(x) + isin(x) + cos(x) - isin(x)2 \ &= cos(x) endalign >