Definition:Cosine Integral Function

Definition

The cosine integral function is the real function $\Ci: \R_{>0} \to \R$ defined as:

$\map \Ci x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {\cos t} t \rd t$

Graph

Also denoted as

The cosine integral function $\map \Ci x$ is also seen denoted $\map {Ci} x$.

Also defined as

Some sources give the cosine integral function as:

$\map \Ci x = \ds -\int_{t \mathop = x}^{t \mathop \to +\infty} \frac {\cos t} t \rd t$

The difference between this and the definition given is trivial, but it is important to check results using this concept to confirm which version is being used.

Also see

• Characterization of Cosine Integral Function

• Definition:Sine Integral Function

• Results about the cosine integral function can be found here.

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {V}$. The Sine and Cosine Integrals
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Miscellaneous Special Functions: Cosine Integral $\ds \map {Ci} x = \int_x^\infty \frac {\cos u} u \rd u$
• 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: Cosine Integral $\ds \map \Ci x = \int_x^\infty \frac {\cos u} u \rd u$

• Weisstein, Eric W. "Cosine Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CosineIntegral.html

which gives the negated version