WOLFRAM

gives the Fresnel auxiliary function TemplateBox[{z}, FresnelF].

Details

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

Examples

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Basic Examples  (4)Summary of the most common use cases

Evaluate numerically:

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Plot over a subset of the reals:

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Plot over a subset of the complexes:

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Series expansion at the origin:

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Scope  (32)Survey of the scope of standard use cases

Numerical Evaluation  (5)

Evaluate to high precision:

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Precision of the output tracks the precision of the input:

Out[2]=2

Evaluate for complex argument:

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Evaluate FresnelF efficiently at high precision:

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Out[2]=2

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

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Or compute average-case statistical intervals using Around:

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Compute the elementwise values of an array:

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Or compute the matrix FresnelF function using MatrixFunction:

Out[2]=2

Specific Values  (3)

Value at a fixed point:

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Values at infinity:

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Out[2]=2

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

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Out[2]=2

Visualization  (2)

Plot the FresnelF function:

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Plot the real part of TemplateBox[{z}, FresnelF]:

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Plot the imaginary part of TemplateBox[{z}, FresnelF]:

Out[2]=2

Function Properties  (9)

FresnelF is defined for all real and complex values:

Out[1]=1
Out[2]=2

Approximate function range of FresnelF:

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FresnelF is an analytic function of x:

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FresnelF is monotonic in a specific range:

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Out[2]=2

FresnelF is not injective:

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Out[2]=2

FresnelF is not surjective:

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Out[2]=2

FresnelF is neither non-negative nor non-positive:

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FresnelF has no singularities or discontinuities:

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Out[2]=2

Neither convex nor concave:

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Differentiation and Integration  (5)

First derivative:

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Higher derivatives:

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Out[2]=2

Indefinite integral of FresnelF:

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More integrals:

Approximation of the definite integral of FresnelF:

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Series Expansions  (4)

Taylor expansion for FresnelF:

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Plot the first three approximations for FresnelF around :

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Taylor expansion for FresnelF at a generic point:

Out[1]=1

Find series expansion at infinity:

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Give the result for an arbitrary symbolic direction :

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Function Identities and Simplifications  (2)

Primary definition:

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Argument simplifications:

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Other Features  (2)

FresnelF threads elementwise over lists and matrices:

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TraditionalForm typesetting:

Applications  (3)Sample problems that can be solved with this function

Interference pattern at the edge of a shadow:

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Plot a clothoid:

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A solution of the timedependent 1D Schrödinger equation for a sudden opening of a shutter:

Check the Schrödinger equation:

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Plot the timedependent solution:

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Neat Examples  (1)Surprising or curious use cases

A generalized helix in terms of Fresnel auxiliary functions:

Out[1]=1
Wolfram Research (2014), FresnelF, Wolfram Language function, https://reference.wolfram.com/language/ref/FresnelF.html.
Wolfram Research (2014), FresnelF, Wolfram Language function, https://reference.wolfram.com/language/ref/FresnelF.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_fresnelf, author="Wolfram Research", title="{FresnelF}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FresnelF.html}", note=[Accessed: 08-May-2025 ]}

@misc{reference.wolfram_2025_fresnelf, author="Wolfram Research", title="{FresnelF}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FresnelF.html}", note=[Accessed: 08-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_fresnelf, organization={Wolfram Research}, title={FresnelF}, year={2014}, url={https://reference.wolfram.com/language/ref/FresnelF.html}, note=[Accessed: 08-May-2025 ]}

@online{reference.wolfram_2025_fresnelf, organization={Wolfram Research}, title={FresnelF}, year={2014}, url={https://reference.wolfram.com/language/ref/FresnelF.html}, note=[Accessed: 08-May-2025 ]}