FresnelF
✖
FresnelF
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
.
- 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
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0bimmcvu-eu0l0o

Plot over a subset of the reals:

https://wolfram.com/xid/0bimmcvu-ftfryb

Plot over a subset of the complexes:

https://wolfram.com/xid/0bimmcvu-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0bimmcvu-2zi51

Scope (32)Survey of the scope of standard use cases
Numerical Evaluation (5)

https://wolfram.com/xid/0bimmcvu-fwug8e

Precision of the output tracks the precision of the input:

https://wolfram.com/xid/0bimmcvu-gctvu8

Evaluate for complex argument:

https://wolfram.com/xid/0bimmcvu-cjypew

Evaluate FresnelF efficiently at high precision:

https://wolfram.com/xid/0bimmcvu-di5gcr


https://wolfram.com/xid/0bimmcvu-bq2c6r

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

https://wolfram.com/xid/0bimmcvu-73ttq


https://wolfram.com/xid/0bimmcvu-lmyeh7


https://wolfram.com/xid/0bimmcvu-i86jgd

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0bimmcvu-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0bimmcvu-thgd2

Or compute the matrix FresnelF function using MatrixFunction:

https://wolfram.com/xid/0bimmcvu-o5jpo

Specific Values (3)

https://wolfram.com/xid/0bimmcvu-cgex6s


https://wolfram.com/xid/0bimmcvu-bdij6w


https://wolfram.com/xid/0bimmcvu-drqkdo

Find a local maximum as a root of :

https://wolfram.com/xid/0bimmcvu-f2hrld


https://wolfram.com/xid/0bimmcvu-ba5463

Visualization (2)
Plot the FresnelF function:

https://wolfram.com/xid/0bimmcvu-ecj8m7


https://wolfram.com/xid/0bimmcvu-ouu484


https://wolfram.com/xid/0bimmcvu-ixzcc6

Function Properties (9)
FresnelF is defined for all real and complex values:

https://wolfram.com/xid/0bimmcvu-cl7ele


https://wolfram.com/xid/0bimmcvu-de3irc

Approximate function range of FresnelF:

https://wolfram.com/xid/0bimmcvu-evf2yr

FresnelF is an analytic function of x:

https://wolfram.com/xid/0bimmcvu-h5x4l2

FresnelF is monotonic in a specific range:

https://wolfram.com/xid/0bimmcvu-g6kynf


https://wolfram.com/xid/0bimmcvu-nlz7s

FresnelF is not injective:

https://wolfram.com/xid/0bimmcvu-gi38d7


https://wolfram.com/xid/0bimmcvu-ctca0g

FresnelF is not surjective:

https://wolfram.com/xid/0bimmcvu-hkqec4


https://wolfram.com/xid/0bimmcvu-hdm869

FresnelF is neither non-negative nor non-positive:

https://wolfram.com/xid/0bimmcvu-84dui

FresnelF has no singularities or discontinuities:

https://wolfram.com/xid/0bimmcvu-mdtl3h


https://wolfram.com/xid/0bimmcvu-mn5jws


https://wolfram.com/xid/0bimmcvu-kdss3

Differentiation and Integration (5)

https://wolfram.com/xid/0bimmcvu-mmas49


https://wolfram.com/xid/0bimmcvu-nfbe0l


https://wolfram.com/xid/0bimmcvu-fxwmfc

Indefinite integral of FresnelF:

https://wolfram.com/xid/0bimmcvu-bponid


https://wolfram.com/xid/0bimmcvu-cf948


https://wolfram.com/xid/0bimmcvu-fcaobo

Approximation of the definite integral of FresnelF:

https://wolfram.com/xid/0bimmcvu-b9jw7l

Series Expansions (4)
Taylor expansion for FresnelF:

https://wolfram.com/xid/0bimmcvu-ewr1h8

Plot the first three approximations for FresnelF around :

https://wolfram.com/xid/0bimmcvu-binhar

Taylor expansion for FresnelF at a generic point:

https://wolfram.com/xid/0bimmcvu-y5gz3

Find series expansion at infinity:

https://wolfram.com/xid/0bimmcvu-eyiujb

Give the result for an arbitrary symbolic direction :

https://wolfram.com/xid/0bimmcvu-hfjwwv

Function Identities and Simplifications (2)
Other Features (2)
FresnelF threads elementwise over lists and matrices:

https://wolfram.com/xid/0bimmcvu-f1ftoj


https://wolfram.com/xid/0bimmcvu-do0xrl

TraditionalForm typesetting:

https://wolfram.com/xid/0bimmcvu-hlustv

Applications (3)Sample problems that can be solved with this function
Interference pattern at the edge of a shadow:

https://wolfram.com/xid/0bimmcvu-bmvkiq


https://wolfram.com/xid/0bimmcvu-fl5keb

A solution of the time‐dependent 1D Schrödinger equation for a sudden opening of a shutter:

https://wolfram.com/xid/0bimmcvu-f01ysn
Check the Schrödinger equation:

https://wolfram.com/xid/0bimmcvu-c4guae

Plot the time‐dependent solution:

https://wolfram.com/xid/0bimmcvu-epgxic

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
]}
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
]}