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

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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  (17)

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]:

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:

Introduced in 2014
 (10.0)