FresnelG

FresnelG[z]

gives the Fresnel auxiliary function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{z}, FresnelG]=(1/2-TemplateBox[{z}, FresnelC]) cos(pi z^2/2)+(1/2-TemplateBox[{z}, FresnelS]) sin(pi z^2/2).
  • FresnelG[z] is an entire function of z with no branch cut discontinuities.
  • For certain special arguments, FresnelG automatically evaluates to exact values.
  • FresnelG can be evaluated to arbitrary numerical precision.
  • FresnelG automatically threads over lists.

Examples

open allclose all

Basic Examples  (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (26)

Numerical Evaluation  (3)

Evaluate numerically to high precision:

Precision of the output tracks the precision of the input:

Evaluate for complex argument:

Evaluate FresnelG efficiently at high precision:

Specific Values  (3)

Value at a fixed point:

Values at infinity:

Find a local maximum as a root of (dTemplateBox[{x}, FresnelG])/(dx)=0:

Visualization  (2)

Plot the FresnelG function:

Plot the real part of TemplateBox[{{x, +, {ⅈ,  , y}}}, FresnelG]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}}, FresnelG]:

Function Properties  (9)

FresnelG is defined for all real and complex values:

Approximate function range of FresnelG:

FresnelG is an analytic function of x:

FresnelG is monotonic in a specific range:

FresnelG is not injective:

FresnelG is not surjective:

FresnelG is neither non-negative nor non-positive:

FresnelG has no singularities or discontinuities:

Neither convex nor concave:

Differentiation and Integration  (3)

First derivative:

Higher derivatives:

Approximation of the definite integral of FresnelG:

Series Expansions  (2)

Taylor expansion for FresnelG:

Plot the first three approximations for FresnelG around :

Taylor expansion for FresnelG at a generic point:

Function Identities and Simplifications  (2)

Primary definition:

Argument simplifications:

Other Features  (2)

FresnelG threads elementwise over lists and matrices:

TraditionalForm typesetting:

Applications  (1)

Interference pattern at the edge of a shadow:

Wolfram Research (2014), FresnelG, Wolfram Language function, https://reference.wolfram.com/language/ref/FresnelG.html.

Text

Wolfram Research (2014), FresnelG, Wolfram Language function, https://reference.wolfram.com/language/ref/FresnelG.html.

BibTeX

@misc{reference.wolfram_2021_fresnelg, author="Wolfram Research", title="{FresnelG}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FresnelG.html}", note=[Accessed: 18-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_fresnelg, organization={Wolfram Research}, title={FresnelG}, year={2014}, url={https://reference.wolfram.com/language/ref/FresnelG.html}, note=[Accessed: 18-October-2021 ]}

CMS

Wolfram Language. 2014. "FresnelG." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FresnelG.html.

APA

Wolfram Language. (2014). FresnelG. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FresnelG.html