differt.em.F

F(z)

Evaluate the transition function at the given points.

The transition function is defined as follows [11, eq. 4.72, p. 184]:

\[F(x) = 2j \sqrt{x} e^{j x} \int\limits_\sqrt{x}^\infty e^{-j u^2} \text{d}u,\]

where \(j^2 = -1\).

As detailed in [11, p. 164], the integral can be expressed in terms of Fresnel integrals (\(C(x)\) and \(S(x)\)), so that:

\[C(x) - j S(x) = \int\limits_\sqrt{x}^\infty e^{-j u^2} \text{d}u.\]

Thus, the transition function can be rewritten as:

\[2j \sqrt{z} e^{j z} \Big(\sqrt{\frac{\pi}{2}}\frac{1 - j}{2} - C(\sqrt{z}) + j S(\sqrt{z})\Big).\]

With Fresnel integrals computed by jax.scipy.special.fresnel.

Parameters:

z (Float[Array, '*batch']) – The array of real points to evaluate.

Return type:

Complex[Array, '*batch']

Returns:

The values of the transition function at the given point.

Examples

The following example reproduces the same plot as in [11, fig. 4.16, p. 185].

>>> from differt.em import F
>>>
>>> x = jnp.logspace(-3, 1, 100)
>>> y = F(x)
>>>
>>> A = jnp.abs(y)  # Amplitude of F(x)
>>> P = jnp.angle(y, deg=True)  # Phase (in deg.) of F(x)
>>>
>>> fig, ax1 = plt.subplots()
>>>
>>> ax1.semilogx(x, A)
>>> ax1.set_xlabel("$x$")
>>> ax1.set_ylabel("Magnitude - solid line")
>>> ax2 = plt.twinx()
>>> ax2.semilogx(x, P, "--")
>>> ax2.set_ylabel("Phase (°) - dashed line")
>>> plt.tight_layout()