Evaluate numerically to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate FresnelS 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 FresnelS function using MatrixFunction:

Value at a fixed point:

Values at infinity:

Find a local maximum as a root of :

Plot the FresnelS function:

Plot the real part of :

Plot the imaginary part of :

FresnelS is defined for all real and complex values:

Approximate function range of FresnelS:

FresnelS is an odd function:

FresnelS is an analytic function of x:

FresnelS is neither non-increasing nor non-decreasing:

FresnelS is not injective:

Not surjective:

FresnelS is neither non-negative nor non-positive:

FresnelS has no singularities or discontinuities:

Neither convex nor concave:

First derivative:

Higher derivatives:

Formula for the derivative:

Indefinite integral of FresnelS:

Definite integral of an odd function over an interval centered at the origin is 0:

More integrals:

Taylor expansion for FresnelS:

Plot the first three approximations for FresnelS around :

General term in the series expansion of FresnelS:

Find series expansion at infinity:

Give the result for an arbitrary symbolic direction :

FresnelS can be applied to power series:

Compute the Laplace transform using LaplaceTransform:

MellinTransform:

Verify an identity relating HypergeometricPFQ to FresnelS:

Argument simplifications:

Integral representation:

Relation to the error function Erf:

FresnelS can be represented as a DifferentialRoot:

FresnelS can be represented in terms of MeijerG:

TraditionalForm formatting: