Fresnel integrals

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1: 7.2 Definitions

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§7.2(iii) Fresnel Integrals

… ► $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. ►

Values at Infinity

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$\underset{x\to \mathrm{\infty }}{lim}C\left(x\right)=\frac{1}{2},$

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§7.2(iv) Auxiliary Functions

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2: Sidebar 7.SB1: Diffraction from a Straightedge

… ►The intensity distribution follows ${|ℱ\left(x\right)|}^2$, where $ℱ$ is the Fresnel integral (See 7.3.4). Fresnel integrals have many applications in optics. …

3: 7.5 Interrelations

§7.5 Interrelations

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7.5.2 $$C\left(z\right)+\mathrm{i}S\left(z\right)=\frac{1}{2}(1+\mathrm{i})-ℱ\left(z\right).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $ℱ\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.6 $${\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}(\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right))=\frac{1}{2}(1\pm \mathrm{i})-(C\left(z\right)\pm \mathrm{i}S\left(z\right)).$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.8 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\mathrm{erf}\zeta .$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{erf}z$: error function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5, §7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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4: 7.4 Symmetry

§7.4 Symmetry

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$C\left(-z\right)=-C\left(z\right),$

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$S\left(-z\right)=-S\left(z\right),$

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$C\left(\mathrm{i}z\right)=\mathrm{i}C\left(z\right),$

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$S\left(\mathrm{i}z\right)=-\mathrm{i}S\left(z\right).$

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5: 7.3 Graphics

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Figure 7.3.3: Fresnel integrals $C\left(x\right)$ and $S\left(x\right)$, $0\le x\le 4$. Magnify

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Figure 7.3.4: ${|ℱ\left(x\right)|}^2$, $-8\le x\le 8$. Fresnel (1818) introduced the integral $ℱ\left(x\right)$ in his study of the interference pattern at the edge of a shadow. He observed that the intensity distribution is given by ${\left|ℱ\left(x\right)\right|}^2$. Magnify

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6: 7.25 Software

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§7.25(iv) $C\left(x\right)$, $S\left(x\right)$, $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in ℝ$

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§7.25(v) $C\left(z\right)$, $S\left(z\right)$, $z\in ℂ$

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§7.25(vi) $ℱ\left(x\right)$, $G\left(x\right)$, $𝖴(x,t)$, $𝖵(x,t)$, $x\in ℝ$

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§7.25(vii) $ℱ\left(z\right)$, $G\left(z\right)$, $z\in ℂ$

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7: 7.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the argument. ►The main functions treated in this chapter are the error function $\mathrm{erf}z$; the complementary error functions $\mathrm{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$; the Voigt functions $𝖴(x,t)$ and $𝖵(x,t)$. ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. …

8: 7.20 Mathematical Applications

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§7.20(ii) Cornu’s Spiral

►Let the set $\{x(t),y(t),t\}$ be defined by $x(t)=C\left(t\right)$, $y(t)=S\left(t\right)$, $t\ge 0$. … ►

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Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by $x=C\left(t\right)$, $y=S\left(t\right)$, $t\in [0,\mathrm{\infty })$. Magnify

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7.20.1 $$\frac{1}{\sigma \sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^x{\mathrm{e}}^{-{(t-m)}^2/(2{\sigma }^2)}dt=\frac{1}{2}\mathrm{erfc}\left(\frac{m-x}{\sigma \sqrt{2}}\right)=Q\left(\frac{m-x}{\sigma }\right)=P\left(\frac{x-m}{\sigma }\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $x$: real variable, $m$: mean and $\sigma$: standard deviation

A&S Ref:

7.1.22 (slightly modified)

Permalink:

http://dlmf.nist.gov/7.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.20(iii), §7.20 and Ch.7

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9: 7.13 Zeros

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§7.13(iii) Zeros of the Fresnel Integrals

… ►Similarly for $S\left(z\right)$. … ►

Table 7.13.3: Complex zeros $x_n+\mathrm{i}{y}_n$ of $C\left(z\right).$

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$n$	$x_n$	$y_n$
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Table 7.13.4: Complex zeros $x_n+\mathrm{i}{y}_n$ of $S\left(z\right)$.

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$n$	$x_n$	$y_n$
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§7.13(iv) Zeros of $ℱ\left(z\right)$

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10: 7.6 Series Expansions

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§7.6(i) Power Series

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7.6.4 $$C\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{(\frac{1}{2}\pi )}^{2n}}{(2n)!(4n+1)}{z}^{4n+1},$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.11

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.6 $$S\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{(\frac{1}{2}\pi )}^{2n+1}}{(2n+1)!(4n+3)}{z}^{4n+3},$$

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Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.13

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.10 $$C\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{𝗃}_{2n}\left(\frac{1}{2}\pi z^2\right),$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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7.6.11 $$S\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{𝗃}_{2n+1}\left(\frac{1}{2}\pi z^2\right).$$

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Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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