FRESNEL

The Fresnel integrals are oscillatory integrals used in diffraction and wave optics. They produce two coupled outputs, sine-like and cosine-like accumulations.

S(z)=\int_0^z \sin\left(\frac{\pi t^2}{2}\right)dt,\qquad C(z)=\int_0^z \cos\left(\frac{\pi t^2}{2}\right)dt

This wrapper evaluates both integrals for a scalar real input and returns a one-row 2D array as [[S, C]].

Excel Usage

=FRESNEL(z)

z (float, required): Real argument for the Fresnel integrals (dimensionless).

Returns (list[list]): One-row array containing Fresnel integrals as [[S, C]].

Example 1: Fresnel integrals at zero

Inputs:

z
0

Excel formula:

=FRESNEL(0)

Expected output:

Result
0	0

Example 2: Fresnel integrals at one half

Inputs:

z
0.5

Excel formula:

=FRESNEL(0.5)

Expected output:

Result
0.0647324	0.492344

Example 3: Fresnel integrals at one

Inputs:

z
1

Excel formula:

=FRESNEL(1)

Expected output:

Result
0.438259	0.779893

Example 4: Fresnel integrals at two

Inputs:

z
2

Excel formula:

=FRESNEL(2)

Expected output:

Result
0.343416	0.488253

Python Code

from scipy.special import fresnel as scipy_fresnel

def fresnel(z):
    """
    Compute Fresnel sine and cosine integrals for a real input.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.fresnel.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        z (float): Real argument for the Fresnel integrals (dimensionless).

    Returns:
        list[list]: One-row array containing Fresnel integrals as [[S, C]].
    """
    try:
        z = float(z)
        s_val, c_val = scipy_fresnel(z)
        return [[float(s_val), float(c_val)]]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

z *

Real argument for the Fresnel integrals (dimensionless).