exojax.special package¶
Submodules¶
exojax.special.erfcx module¶
- exojax.special.erfcx.erfcx(float) based on Shepherd and Laframboise (1981)¶
Scaled complementary error function exp(-x*x) erfc(x)
Args: x: should be larger than -9.3
- Returns
erfcx(x)
- Return type
jnp.array
Note
We acknowledge the post in stack overflow (https://stackoverflow.com/questions/39777360/accurate-computation-of-scaled-complementary-error-function-erfcx).
exojax.special.expn module¶
- exojax.special.expn.E1(x)¶
Abramowitz Stegun (1970) approximation of the exponential integral of the first order, E1.
- Parameters
x – input
- Returns
The exponential integral of the first order, E1(x)
exojax.special.faddeeva module¶
Faddeeva (wofz= w of z) functions (real and imag parts), an asymptotic form of wofz.
Note
We adopt ncut=27 as the summation of n in Algorithm 916. For Sigma1, we use ncut=8 because higher order than 9 does not affect the results.
- exojax.special.faddeeva.asymptotic_wofz(x, y)¶
Asymptotic representation of wofz (Faddeeva) function 1 for |z|**2 > 112 (for e = 10e-6)
See Zaghloul (2018) arxiv:1806.01656
- Parameters
x – real x
y – real y
- Returns
wofz(x+iy)
- Return type
jnp.array (complex)
- exojax.special.faddeeva.faddeeva_sigma1(exx, y2)¶
- exojax.special.faddeeva.h_bwd(res, g)¶
backward.
Note
V=Real(wofz), L=Imag(wofz)
- Parameters
res – res from h_fwd
g – g
- Returns
g* partial_x h(x,y), g* partial_y h(x,y)
- Return type
jnp.array, jnp.array
- exojax.special.faddeeva.h_fwd(x, y)¶
- exojax.special.faddeeva.imwofz(x, y)¶
Imaginary part of wofz (Faddeeva) function based on Algorithm 916.
We apply a=0.5 for Algorithm 916.
- Parameters
x – x < ncut/2
y –
- Returns
Imag(wofz(x+iy))
- Return type
jnp.array
- exojax.special.faddeeva.rewofz(x, y)¶
Real part of wofz (Faddeeva) function based on Algorithm 916.
We apply a=0.5 for Algorithm 916.
- Parameters
x – x < ncut/2
y –
- Returns
Real(wofz(x+iy))
- Return type
jnp.array
exojax.special.j0 module¶
- exojax.special.j0.j0(x)¶
Bessel function of the 1st kind, order=0.
- Parameters
x – x
- Returns
J0