gives the Fresnel integral .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- FresnelS[z] is given by .
- FresnelS[z] is an entire function of z with no branch cut discontinuities.
- For certain special arguments, FresnelS automatically evaluates to exact values.
- FresnelS can be evaluated to arbitrary numerical precision.
- FresnelS automatically threads over lists.
Examplesopen allclose all
Basic Examples (5)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Numerical Evaluation (4)
Evaluate numerically to high precision:
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate FresnelS efficiently at high precision:
FresnelS threads elementwise over lists and matrices:
Specific Values (3)
Value at a fixed point:
Values at infinity:
Find a local maximum as a root of :
Plot the FresnelS function:
Plot the real part of :
Plot the imaginary part of :
Function Properties (3)
FresnelS is defined for all real and complex values:
Approximate function range of FresnelS:
FresnelS is an odd function:
Formula for the derivative:
Indefinite integral of FresnelS:
Definite integral of an odd function over an interval centered at the origin is 0:
Series Expansions (5)
Taylor expansion for FresnelS:
Plot the first three approximations for FresnelS around :
General term in the series expansion of FresnelS:
Find series expansion at infinity:
Give the result for an arbitrary symbolic direction :
FresnelS can be applied to power series:
Function Identities and Simplifications (2)
Function Representations (5)
Intensity of a wave diffracted by a half‐plane:
Plot a Cornu spiral:
A solution of the time‐dependent 1D Schrödinger equation for a sudden opening of a shutter:
Check the Schrödinger equation:
Plot the time‐dependent solution:
Plot of FresnelS along a circle in the complex plane:
Properties & Relations (6)
Use FullSimplify to simplify expressions containing Fresnel integrals:
Find a numerical root:
Obtain FresnelS from integrals and sums:
Solve a differential equation:
Calculate the Wronskian:
Compare with Wronskian:
Possible Issues (2)
FresnelS can take large values for moderate‐size arguments:
Different convention can sometimes be seen in books: