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 Subject: physics/math Category: Science > Physics Asked by: mathwhiz-ga List Price: \$25.00 Posted: 07 Nov 2002 15:37 PST Expires: 07 Dec 2002 15:37 PST Question ID: 102241
 ```This is an extra credit math problem that i really want to solve. I need the proof that e^i*O = i*sin0 + cosO. where O=theta, and also explain why this discovery is the basis of most physics principles in mechanics. I need this answer BY tomorrow 11/8/2002 I really would appreciate any help that could be given to me.``` Request for Question Clarification by rbnn-ga on 07 Nov 2002 16:25 PST ```Hi there, Well, can I use a little calculus? It's easiest to show this with a little bit of calculus actually.```
 ```Thanks for your question. There are many ways of answering this kind of question and many levels of detail into which one can delve. I will give one answer here, and, if you like, just let me know if there is any more information or greater clarification you would like over any particular point and I would be happy to answer it (use the "Request Clarification" button to ask for more clarification). The formula i theta e = cos (theta) + i sin(theta) is indeed one of the most important and beautiful formulas in mathematics. Indeed, if we substitute the value pi for theta, we get i pi e = cos (pi) + i sin(pi) = -1 + i 0 = -1 (Remember we are measuring theta in radians, so that 2pi radians = 360 degrees.) I remember when I first saw this formula as a kid how impressed I was. It was one of the things that made me more curious about higher mathematics actually. This formula, "Euler's identity" relating three fundamental constants in mathematics (e, i, and pi --- four if you count -1) has been called "the most remarkable formula in the world. "Euler's identity" http://www.wikipedia.org/wiki/The_most_remarkable_formula_in_the_world points to "Most remarkable theorem in the world" . The goal of this short essay is first, to prove the formula; and second, to discuss its relevance to mechanics. ------------------- PROVING THE FORMULA In order to prove the formula, it is necessary first to define "e" and second to define the notion of raising e to the power of a complex number . As is virtually always the case for this sort of thing, the definitions of the main terms have undergone considerable modification over the centuries. I will give a somewhat more modern treatment here. However, I will try and motivate the definitions a little bit. One way to define e is to start by defining the exponential function x e One can define x 2 3 n e = 1 + x + x + x + ... + x + ... --- --- --- 2! 3! n! Here, n!, or n factorial, denotes 1*2* ... *n, the product of the first n numbers. 0! is defined to be 1. More compactly, this is sometimes written: x j=infinity j e = Sum x j=0 ---- j! Now, in order to prove Euler's formula, we need to know a few facts about Taylor series. Every real function (well, every "well-behaved", ) function f(x) can be written as a an infinite-degree polynomial in f: j=infinity j f(x) = Sum a x j=0 j Here x is a real number and each of the a's are also real numbers. The value of each of the a_j's can be computed using calculus, using the so-called Taylor formula. (n) a = f (0) j ------ n! where (n) f (x) is the value of the n'th derivative of f at 0. ----------Side topic: derivatives-------------- I'm not sure how much about calculus you know, so I will briefly define a derivative. Given a nice real function f, we can compute the derivative of f at a point x. It is written f'(x) or (1) f (x) in the notation above. The derivative is the *rate of change* of f at a particular point. Formally, f(x+h)-f(x) f'(x) = lim ------------- h->0 x-h For examples, suppose a car is moving in straight line to the right. Let f(t) be the distance of the car in feet from its starting location after t seconds. Then f'(t) will equal the speed, in feet-per-second of the car at time t. Also, f''(t) will equal the acceleration of the time at time t. Here are three facts about derivatives: n 1. If f(x) = x , for some positive integer n, then n-1 f'(x) = n x 2. If f(x)=u(x) + v(x), then f'(x)=u'(x)+v'(x) . 3. If f(x) = sin(x) then f'(x)=cos(x) . 4. If f(x)=cos(x) then f'(x)= -sin(x) . Now, using facts 3 and 4, we can derive the Taylor series for sin(x) and cos(x) . That's because the derivatives of, say, sin(x) keep repeating, we get sin(x) has derivative of: cos(x) which has derivative: -sin(x) with derivative: -cos(x) with derivative: sin(x) and we are back to the beginning. Anyway, when one carries out the computation for the Taylor series of sin and cos, one gets this result: sin(x) = 1 3 1 5 1 7 x - --- x + ---- x - ---- x + ... 3! 5! 7! That is sin(x) = j=infinity j 1 (2j+1) Sum (-1) * -------- x j=0 (2j+1)! Similarly, the Taylor series for cos(x) is: 1 2 1 5 1 7 1- --- x + ---- x - ---- x + ... 2! 5! 7! with general term of: j=infinity j 1 (2j) Sum (-1) * -------- x j=0 (2j)! Now, it is easy to prove Euler's formula from the general expression for e using the Taylor formula for e above. We use the fact that i*i= -1. So, i*t e = 2 3 4 5 6 1+(i t)+(i t)+ (i t) + (i t) + (i t) + (i t) + ... --- --- ---- --- --- 2! 3! 4! 5! 6! n n n If we examine a numerator ( i t ) = i t = n t if 4 divides n evenly n i t if n divided by 4 leaves a remainder of 1 n - t if n divided by 4 leaves a remainder of 2 n -i t if n divided by 4 leaves a remainder of 3 it So, when you plug this back into the equation for e you get: it e 2 3 4 5 = t t t t 1 + it - --- - i ---- + ----- + i----- ... 2! 3! 4! 5! Finally, for the big finish, if you collect all the imaginary terms together, you find that they equal i sin(t) using the Taylor series for sin. it Similarly the sum of all the real terms in the expression for e is cos(t) . So it e = cos(t) + i sin(t) . This is the proof. ------Importance of exponential function to mechanics----- z The importance of the exponential function (that is, e ) in mechanics, and indeed in all of physics, lies in the following crucial property of x e Namely, its derivative equals itself. x That is, if f(x) = e , then f'(x) = f(x) . x The only other function with this property is multiple of f, like 2*e . x But e also satisfies many other nice properties like 0 e =1 x+y x y e = e e It's the only function with these properties whose derivative is equal to itself. The reason that this is so important is that the laws of mechanics are all differential equations. A differential equation is some expression that relates the values of a function to the values of its derivatives. Specifically, Newton's laws are differential equations (it's not coincidence that Newton was involved so much in both the development of physics and of calculus). For example, consider Newton's law: F = m a , Force is mass times acceleration. Well, we saw above that acceleration is the second derivative of position , so solving differential equations is fundamental to solving mechanics problems. The exponential function is crucial to solving differential equations because it is an identity for them, in the same way that the number 0 is crucial for solving addition problems, or the number 1 for solving multiplication problems. Any time there is oscillatory motion in mechanics, in particular, the exponential function arises. But, when we solve differential equations in terms of complex exponentials, we have to find some way to interpret the results in terms of real numbers. The Euler equation here gives us that formula. Let's look at a simple explanation for the power of the exponential. Remember in trigonometry where one learned some complex formula for sin(x+y) . Well, now it is trivial to derive, since sin(x+y) = i(x+y) Im( e ) ix iy = Im (e e ) = Im ( (cos x + i sin x) (cos y + i sin y) = cos x sin y + sin x cos y . Similarly, one can get really quick formulas for sin(5 x) for example, that would be a major hassle to derive without using exponential. By the way, it is worth noting that in quantum mechanics just about everything is complex: according to quantum mechanics every particle has an associated (complex) wave function that pretty much determines everything about that particle. So in quantum mechanics, exponentials are even more important. LINKS ------ Brief discussion of Euler's formula: http://www.ma.iup.edu/projects/CalcDEMma/complex/complex4.html Euler formula again: http://www.wikipedia.org/wiki/Eulers_formula_in_complex_analysis "Euler's identity" http://www.wikipedia.org/wiki/The_most_remarkable_formula_in_the_world points to "Most remarkable theorem in the world" "History of the exponential function", supposedly proved by Cotes in 1714, and in 1740-1748, Euler showed the series expansion. Plot of complex exponential function: http://www.kfunigraz.ac.at/imawww/vqm/pages/complex/05_exp.html 10000 digits of e here: http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/e_10000.html (Always useful for increasing the page count). Sin series and cos series: http://www.ms.uky.edu/~carl/ma330/sin/sin1.html```
 ```One interesting relationship which fascinates me is that i to the power of i is actually a real number. I think it is: i^i = e^(-pi/2)```
 ```alan0-ga: Thank you for your comment. http://westview.tdsb.on.ca/Mathematics/data/complex.html In general, when dealing with complex powers one must be fairly careful with "branch cuts": making sure that we account for all possible "interpretations". For example (-1)^{1/2} has two solutions, not one. That said, since cos(pi/2)=0 and sin(pi/2) = 1 , we can write: i^i = (0*cos(pi/2)+ i*sin(pi/2))^i = (exp(i*pi/2))^i = exp(i*i*pi/2) =exp(-pi/2) . This is quite cute, thanks. The thing is, we can also write i = exp(-i 5*pi/2) and get a different answer. See the URL: http://westview.tdsb.on.ca/Mathematics/data/complex.html for further details.```