Ascending series for C and S were given by K. W. Knockenhauer, and are readily investigated. Integrating by parts, we find
| C + iS = ∫ | v | e | i·½πv² | dv = e | i·½πv² | · v − 1⁄3 iπ ∫ | v | e | i·½πv² | dv³; |
| 0 | 0 |
and, by continuing this process,
| C + iS = e | i·½πv² | { v − | iπ | v³ + | iπ | iπ | v5 − | iπ | iπ | iπ | v7 + ... }. |
| 3 | 3 | 5 | 3 | 5 | 7 |
By separation of real and imaginary parts,
| C = M cos ½πv² − N sin ½πv² | } (6), |
| S = M sin ½πv² − N cos ½πv² |
where
| M = | v | − | π²v5 | + | π4v9 | − ... (7), |
| 1 | 3·5 | 3·5·7·9 |
| N = | πv³ | − | π3v7 | + | π5v11 | ... (8). |
| 1·3 | 1·3·5·7 | 1·3·5·7·9·11 |
These series are convergent for all values of v, but are practically useful only when v is small.