Sine Integral Function is Odd

Theorem

$\map \Si {-x} = -\map \Si x$

where:

$\Si$ denotes the sine integral function

$x$ is a real number.

Proof

\(\ds \map \Si {-x}\)

\(=\)

\(\ds \int_0^{-x} \frac {\sin u} u \rd u\)

Definition of Sine Integral Function

\(\ds \)

\(=\)

\(\ds -\int_0^x \frac {\map \sin {-u} } {-u} \rd u\)

substituting $u \mapsto -u$

\(\ds \)

\(=\)

\(\ds -\int_0^x \frac {\sin u} u \rd u\)

Sine Function is Odd

\(\ds \)

\(=\)

\(\ds -\map \Si x\)

Definition of Sine Integral Function

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Miscellaneous Special Functions: Sine Integral $\ds \map {Si} x = \int_0^x \frac {\sin u} u \rd u$: $35.13$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 36$: Miscellaneous and Riemann Zeta Functions: Sine Integral $\ds \map \Si x = \int_0^x \frac {\sin u} u \rd u$: $36.13.$