Definition:Fresnel Integral/Cosine

Definition

The Fresnel cosine integral function is the real function $\operatorname C: \R \to \R$ defined by:

$\ds \map {\operatorname C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$

Asymptotic Expansion for Fresnel Cosine Integral Function

\(\ds \map {\operatorname C} x\)

\(\sim\)

\(\ds \frac 1 2 + \frac 1 {\sqrt {2 \pi} } \paren {\map \sin {x^2} \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1}!} {2^{3 n} n! x^{4 n + 1} } } - \frac 1 2 \map \cos {x^2} \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1}!} {2^{3 n} n! x^{4 n + 3} } } }\)

\(\ds \)

\(=\)

\(\ds \frac 1 2 + \frac 1 {\sqrt {2 \pi} } \paren {\map \sin {x^2} \paren {\frac 1 x - \frac {1 \times 3} {2^2 x^5} + \frac {1 \times 3 \times 5 \times 7} {2^4 x^9} - \ldots} - \map \cos {x^2} \paren {\frac 1 {2 x^3} - \frac {1 \times 3 \times 5} {2^3 x^7} + \ldots} }\)

Also defined as

The Fresnel cosine integral function can also be seen defined as:

$\ds \map {\operatorname C} x = \int_0^x \cos u^2 \rd u$

In the context of physics, the Fresnel cosine integral function is most often defined as:

$\ds \map {\operatorname C} x = \int_0^x \map \cos {\dfrac {\pi u^2} 2} \rd u$

Also see

• Definition:Fresnel Sine Integral Function

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

Source of Name

This entry was named for Augustin-Jean Fresnel.

Historical Note

The Fresnel integrals were used by Augustin-Jean Fresnel to analyse the diffraction of light.

Linguistic Note

The eponym Fresnel is pronounced fre-nell.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Fresnel Cosine Integral $\ds \map {\operatorname C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$

• Weisstein, Eric W. "Fresnel Integrals." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FresnelIntegrals.html