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 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.