# FresnelG

FresnelG[z]

gives the Fresnel auxiliary function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• 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.
• FresnelG can be used with Interval and CenteredInterval objects. »

# 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(32)

### Numerical Evaluation(5)

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:

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

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix FresnelG function using MatrixFunction:

### Specific Values(3)

Value at a fixed point:

Values at infinity:

Find a local maximum as a root of :

### Visualization(2)

Plot the FresnelG function:

Plot the real part of :

Plot the imaginary part of :

### 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(5)

First derivative:

Higher derivatives:

Indefinite integral of FresnelG:

More integrals:

Approximation of the definite integral of FresnelG:

### Series Expansions(4)

Taylor expansion for FresnelG:

Plot the first three approximations for FresnelG around :

Taylor expansion for FresnelG 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)

FresnelG threads elementwise over lists and matrices:

## 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), 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.

#### 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

#### BibTeX

@misc{reference.wolfram_2024_fresnelg, author="Wolfram Research", title="{FresnelG}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FresnelG.html}", note=[Accessed: 17-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_fresnelg, organization={Wolfram Research}, title={FresnelG}, year={2014}, url={https://reference.wolfram.com/language/ref/FresnelG.html}, note=[Accessed: 17-September-2024 ]}