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


  • 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. »


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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  (29)

Numerical Evaluation  (4)

Evaluate to high precision:

Precision of the output tracks the precision of the input:

Evaluate for complex argument:

Evaluate FresnelF efficiently at high precision:

FresnelF can be used with Interval and CenteredInterval objects:

Specific Values  (3)

Value at a fixed point:

Values at infinity:

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

Visualization  (2)

Plot the FresnelF function:

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

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

Function Properties  (9)

FresnelF is defined for all real and complex values:

Approximate function range of FresnelF:

FresnelF is an analytic function of x:

FresnelF is monotonic in a specific range:

FresnelF is not injective:

FresnelF is not surjective:

FresnelF is neither non-negative nor non-positive:

FresnelF has no singularities or discontinuities:

Neither convex nor concave:

Differentiation and Integration  (3)

First derivative:

Higher derivatives:

Approximation of the definite integral of FresnelF:

Series Expansions  (4)

Taylor expansion for FresnelF:

Plot the first three approximations for FresnelF around :

Taylor expansion for FresnelF at a generic point:

Find series expansion at infinity:

Give the result for an arbitrary symbolic direction :

Function Identities and Simplifications  (2)

Primary definition:

Argument simplifications:

Other Features  (2)

FresnelF threads elementwise over lists and matrices:

TraditionalForm typesetting:

Applications  (3)

Interference pattern at the edge of a shadow:

Plot a clothoid:

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

Check the Schrödinger equation:

Plot the timedependent solution:

Neat Examples  (1)

A generalized helix in terms of Fresnel auxiliary functions:

Wolfram Research (2014), FresnelF, Wolfram Language function,


Wolfram Research (2014), FresnelF, Wolfram Language function,


Wolfram Language. 2014. "FresnelF." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2014). FresnelF. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_fresnelf, author="Wolfram Research", title="{FresnelF}", year="2014", howpublished="\url{}", note=[Accessed: 23-May-2024 ]}


@online{reference.wolfram_2024_fresnelf, organization={Wolfram Research}, title={FresnelF}, year={2014}, url={}, note=[Accessed: 23-May-2024 ]}