7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.2 Definitions 7.4 Symmetry

§7.3 Graphics

Permalink:: http://dlmf.nist.gov/7.3

§7.3(i) Real Variable
§7.3(ii) Complex Variable

§7.3(i) Real Variable

Notes:: These graphs were produced at NIST.
Permalink:: http://dlmf.nist.gov/7.3.i

Figure 7.3.1: Complementary error functions $\mathop{\mathrm{erfc}\/}\nolimits x$ and $\mathop{\mathrm{erfc}\/}\nolimits\!\left(10x\right)$ , $-3\leq x\leq 3$ .

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function and : real variable
Keywords:: error functions
Permalink:: http://dlmf.nist.gov/7.3.F1
Encodings:: pdf, png

Figure 7.3.2: Dawson’s integral $\mathop{F\/}\nolimits\!\left(x\right)$ , $-3.5\leq x\leq 3.5$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral and : real variable
Keywords:: Dawson’s integral
Permalink:: http://dlmf.nist.gov/7.3.F2
Encodings:: pdf, png

Figure 7.3.3: Fresnel integrals $\mathop{C\/}\nolimits\!\left(x\right)$ and $\mathop{S\/}\nolimits\!\left(x\right)$ , $0\leq x\leq 4$ .

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral and : real variable
Keywords:: Fresnel integrals, diffraction of light
Permalink:: http://dlmf.nist.gov/7.3.F3
Encodings:: pdf, png

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