In order to explain Euler's formula we must first address the more basic question,"What does e^(i*theta) mean?" Surprisingly,many authors answer this by defining e^(i*theta),out of the blue,to be (cos(theta)+i*sin(theta))! This gambit is logically unimpeachable,but it is also a low blow to Euler,reducing one of his greatest achievements to a mere tautology. We will therefore give two heuristic arguments in support of(10);deeper arguments will emerge in later chapters.
Recall the basic fact that e^x is its own derivative: de^x/dx = e^x. This is actually a defining property,that is,if df(x)/dx = f(x),and f(0)= 1,then f(x)= e^x. Similarly,if k is a real constant,then e^kx may be defined by the property df(x)/dx = kf(x). To extend the action of the ordinary exponential function e^x from real values of x to imaginary ones,let us cling to this property by insisting that it remain true if k=i,so that
de^it/dt = ie^it (11)
We have used the letter t instead of x because we will now think of the variable as being time. We are used to thinking of the derivative of a real function as the slope of the tangent to the graph of the function,but how are we to understand the derivative in the above equation?
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