Exponential and Trigonometric Functions: Explanation

To calculate et, we can use the following formula:

, where .

Note: This is a special case of the general formula for powers given earlier.  As before, since these are infinite sums, these sums get closer and closer to the exact values, as we include more and more terms in each sum.  

We can use this formula to calculate et i and compute its real and complex parts:

That is, the real part of et i is given by the formula , while its complex part is given by .

We can actually use basic rules of exponents and algebra to derive better and better approximate formulas for et.  For example, we can see from the graphs in the following Example that et » 1 + t:

Notice how the line, 1 + t, is always below the exponential.  Looking at the formula given earlier, this makes sense, since it is missing all the higher power terms. 

Using rules of exponents, et = (et/2)2, we can get an improved approximation: et » (1 + t/2)2 = 1 + t + t2/4.  Repeating this process gives an even better approximation: et » (1 + t/2 + (t/2)2/4)2 = 1 + t + 3t2/8 + t3/16 + t4/256.  We can see this in the following Example:

You can see how, by adding more and more terms, we are getting closer and closer to the exact formula.  Note: We are simply looking at the polynomials ; from the discussion on the number e, it should not be surprising that this expression converges to ex as n grows large.

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