Expressions suitable for discussion when v is large were obtained by L. P. Gilbert (Mem. cour. de l’Acad. de Bruxelles, 31, p. 1). Taking
½πv² = u (9),
we may write
| C + iS = | 1 | ∫ | u | eiudu | (10). |
| √(2π) | 0 | √u |
Again, by a known formula,
| 1 | = | 1 | ∫ | ∞ | e−uxdx | (11). |
| √ u | √π | 0 | √x |
Substituting this in (10), and inverting the order of integration, we get
| C + iS = | 1 | ∫ | ∞ | dx | ∫ | u | e | u(i − x) | dx = | 1 | ∫ | ∞ | dx | eu(i − x) − 1 | (12). |
| π√2 | 0 | √x | 0 | π√2 | √x | i − x |
Thus, if we take
| G = | 1 | ∫ | ∞ | e−ux √x · dx | , H = | 1 | ∫ | ∞ | e−ux dx | (13), |
| π√2 | 0 | 1 + x² | π√2 | 0 | √x · (1 + x²) |