The Fourier series of §20.2(i) usually converge rapidly because of the factors or , and provide a convenient way of calculating values of . Similarly, their -differentiated forms provide a convenient way of calculating the corresponding derivatives. For instance, the first three terms of (20.2.1) give the value of () to 12 decimal places.
For values of near 1 the transformations of §20.7(viii) can be used to replace with a value that has a larger imaginary part and hence a smaller value of . For instance, to find we use (20.7.32) with , . Then and . Hence the first term of the series (20.2.3) for suffices for most purposes. In theory, starting from any value of , a finite number of applications of the transformations and will result in a value of with ; see §23.18. In practice a value with, say, , , is found quickly and is satisfactory for numerical evaluation.