− 1⁄3 (S3 − 1) x3 − 1⁄5 (S5 − 1) x5 − ...,
where
| S2r+1 = 1 + | 1 | + | 1 | + ..., |
| 22r+1 | 32r+1 |
and the series for log Γ(1 + x) converges when x lies between −1 and 1.
52. Definite integrals.Definite integrals can sometimes be evaluated when the limits of integration are some particular numbers, although the corresponding indefinite integrals cannot be found. For example, we have the result
∫10 (1 − x2)−1/2 log x dx = −½ π log 2,
although the indefinite integral of (1 − x2)−1/2 log x cannot be found. Numbers of definite integrals are expressible in terms of the transcendental functions mentioned in § 50 or in terms of Gamma functions. For the calculation of definite integrals we have the following methods:—
| (i.) | Differentiation with respect to a parameter. |
| (ii.) | Integration with respect to a parameter. |
| (iii.) | Expansion in infinite series and integration term by term. |
| (iv.) | Contour integration. |
The first three methods involve an interchange of the order of two limiting operations, and they are valid only when the functions satisfy certain conditions of continuity, or, in case the limits of integration are infinite, when the functions tend to zero at infinite distances in a sufficiently high order (see [Function]). The method of contour integration involves the introduction of complex variables (see [Function]: § Complex Variables).
A few results are added