Very often the first thing we do when we calculate the fourier series of a function is to express cosnt or sinnt in their complex forms (eint+ e-int)/2 and (eint- e-int)/2i. Since the difference between cosnt and sinnt is in any case only one of phase, algebra is simplified if we express the Fourier series in terms of eint.
Notation: in what follows, a symbol will be complex if it has a line over it, e.g.
The fourier series of a function y(t) is written:
Now we know that eint = cosnt + sinnt, so we can expand the Fourier series of y(t) as follows:
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The third term, 
, could be rewritten 
except that if we are going to do this then strictly speaking in order to keep our annotation correct we should define that a-n = an and b-n.= bn . But an + ibn and an - ibn could in any case both be more conveniently expressed by a single complex paramater, so the following two definitions are made:
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for all n>0 |
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for all n>0 |
We can now express y(t) as follows:
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y(t) |
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If we allow ourselves the further definition that 
then we can put all three terms under one summation:
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y(t) |
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This is called the complex Fourier series. Notice that the ‘=’ sign in this expression really does mean ‘equals’, not ‘equals real part of..’.
The complex Fourier series is indeed algebraically more compact than the original, but is there also a compact expression for the calculation of 
?
The expressions derived for the calculation of an and bn are:
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and |
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The definition for was that:
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for n > 0 |
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for n > 0 |
The same expression can be shown to hold for n = 0, and n < 0, so now instead of having to calculate an and bn separately, we can calculate from one compact algebraic equation.
The complex Fourier series is so much simpler than the real Fourier series that even when people want the real Fourier series, they often begin by calculating , then use the result to find an and bn. This can be done by combining our definitions for and 
:
