Euler EquationDate: 01/21/97 at 23:45:40 From: Alan Davis Subject: formula for e I am sure that e^(pi*i) = -1 has no meaning, but I'd like to follow the steps. Date: 01/22/97 at 18:54:58 From: Doctor Mitteldorf Subject: Re: formula for e Dear Alan, e^i*pi = -1 is a special case of the Euler equation e^i*x = cos(x) + i*sin(x) This equation, it turns out, is enormously interesting and useful. It's not arbitrary, in the sense that once you decide to define sqrt(-1) = i, the Euler equation must be true. Here are some of the things you can do to attach some meaning in your own mind to the Euler equation: 1. Write down the infinite series e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... Write down the series for sin(x) and cos(x) sin(x) = x - x^3/3! + x^5/5! + ... cos(x) = 1 - x^2/2! + x^4/4! + ... Use them to verify the Euler equation. 2. Use the trig formulas for cos(2x) and sin(2x) to find cos(2x) + i*sin(2x) Now, if this function (for e) is really an exponential, it should be growing exponentially. In other words, when you take the function with 2x as argument, you should just get the square of what you get when x is the argument. When you take the function of 3x, you should get the cube of the function of x. Does the function cos(x)+i*sin(x) have this property? Test it and see. 3. If you know a little calculus, you can try differentiating the Euler equation to see that you get a consistent answer. There are branches of physics where people are constantly using the Euler equation to go back and forth between the trigonometric and the exponential version of a formula, because first one version then the other turns out to be easier to work with. To them, the Euler equation has become second nature. -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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