def test_plasma_dispersion_func_symmetry(self, w):
r"""Test plasma_dispersion_func against its symmetry properties"""
# The two symmetry properties of the plasma dispersion function
# are taken from the bottom of page 30 of "NRL Plasma Formulary"
# by A.S. Richardson (2019)
Z_of_wconj = plasma_dispersion_func(w.conjugate())
minusZ_of_minuswconj = -(plasma_dispersion_func(-w).conjugate())
assert np.isclose(
Z_of_wconj, minusZ_of_minuswconj, atol=0, rtol=1e-15), (
"The symmetry property of the plasma dispersion function that "
f"Z(w*) == -[Z(-w)]* is not met for w = {w}. Instead, "
f"plasma_dispersion_func({w.conjugate()}) = {Z_of_wconj} "
f"whereas -plasma_dispersion_func({-w}).conjugate() = "
f"{minusZ_of_minuswconj}. "
"The difference between Z(w*) and -[Z(-w)]* is "
f"{Z_of_wconj - minusZ_of_minuswconj}.")
if w.imag > 0:
should_equal_Z_of_wconj = (plasma_dispersion_func(
w)).conjugate() + 2j * np.sqrt(π) * np.exp(-(w.conjugate()**2))
assert np.isclose(
Z_of_wconj, should_equal_Z_of_wconj, rtol=1e-13
), ("The symmetry property of the plasma dispersion function that "
"Z(w*) = Z(w)* + 2j * sqrt(pi) * exp[-(w*)**2] for Im(w) > 0 "
f"is not met for w = {w}. The value of "
f"plasma_dispersion_func({w.conjugate()}) is {Z_of_wconj}, "
f"which is different from {should_equal_Z_of_wconj}. "
"The difference between these two results is "
f"{Z_of_wconj - should_equal_Z_of_wconj}.")
def test_plasma_dispersion_func_errors(self, w, expected_error):
"""Test errors that should be raised by plasma_dispersion_func."""
with pytest.raises(expected_error):
plasma_dispersion_func(w)
pytest.fail(f"plasma_dispersion_func({w}) did not raise "
f"{expected_error.__name__} as expected.")
def test_plasma_dispersion_func_roots(self):
"""Test roots of the plasma dispersion function."""
# The first five roots of the plasma dispersion function are given
# on page 402 of Swanson (2003), with some roundoff or truncation
# error in the final decimal point. These roots were found to
# higher precision using mpmath.findroot.
roots = np.array(
[
1.991_466_842_833_879_6 - 1.354_810_128_112_006_2j,
2.691_149_024_251_438_8 - 2.177_044_906_089_615_9j,
3.235_330_868_352_816_5 - 2.784_387_613_230_428_2j,
3.697_309_702_468_468_4 - 3.287_410_789_389_848_6j,
4.106_107_284_682_632_1 - 3.725_948_719_445_790_4j,
],
dtype=np.complex128,
)
for root in roots:
Z_at_root = plasma_dispersion_func(root)
assert np.isclose(Z_at_root, 0 + 0j, atol=1e-15 * (1 + 1j)), (
"A root of the plasma dispersion function is expected at w = "
f"{root}, but plasma_dispersion_func({root}) is equal to "
f"{Z_at_root} instead of {0j}.")
def test_plasma_dispersion_func(w, expected):
r"""Test plasma_dispersion_func against tabulated results and
known symmetry properties."""
# Many of the tabulated results originally came from the book
# entitled "The Plasma Dispersion Function: The Hilbert Transform
# of the Gaussian" by B. D. Fried and S. D. Conte (1961). The two
# symmetry properties of the plasma dispersion function are at the
# bottom of page 2 of this book.
Z_of_w = plasma_dispersion_func(w)
assert np.isclose(Z_of_w, expected, atol=1e-12 * (1 + 1j), rtol=1e-12), (
f"plasma_dispersion_func({w}) equals {Z_of_w} instead of the "
f"expected approximate result of {expected}. The difference between "
f"the actual and expected results is {Z_of_w - expected}.")
Z_of_wconj = plasma_dispersion_func(w.conjugate())
minusZ_of_minuswconj = -(plasma_dispersion_func(-w).conjugate())
assert np.isclose(Z_of_wconj, minusZ_of_minuswconj, atol=0, rtol=1e-15), (
"The symmetry property of the plasma dispersion function that "
f"Z(w*) == -[Z(-w)]* is not met for w = {w}. Instead, "
f"plasma_dispersion_func({w.conjugate()}) = {Z_of_wconj} "
f"whereas -plasma_dispersion_func({-w}).conjugate() = "
f"{minusZ_of_minuswconj}. "
"The difference between Z(w*) and -[Z(-w)]* is "
f"{Z_of_wconj - minusZ_of_minuswconj}.")
if w.imag > 0:
should_equal_Z_of_wconj = (plasma_dispersion_func(
w)).conjugate() + 2j * np.sqrt(π) * np.exp(-(w.conjugate()**2))
assert np.isclose(Z_of_wconj, should_equal_Z_of_wconj, rtol=1e-14), (
"The symmetry property of the plasma dispersion function that "
"Z(w*) = Z(w) + 2j * sqrt(pi) * exp[-(w*)**2] for Im(w) > 0 "
f"is not met for w = {w}. The value of "
f"plasma_dispersion_func({w.conjugate()}) is {Z_of_wconj}, "
f"which is different from {should_equal_Z_of_wconj}. "
"The difference between these two results is "
f"{Z_of_wconj - should_equal_Z_of_wconj}.")
def test_plasma_dispersion_func_deriv_characterization(self, w):
r"""Test plasma_dispersion_func_deriv against an exact relationship."""
# The exact analytical relationship comes from the bottom of
# page 3 of Fried & Conte (1961).
Z = plasma_dispersion_func(w)
Z_deriv = plasma_dispersion_func_deriv(w)
Z_deriv_characterization = -2 * (1 + w * Z)
assert np.isclose(Z_deriv, Z_deriv_characterization, rtol=1e-15), (
f"The relationship that Z'(w) = -2 * [1 + w * Z(w)] is not "
f"met for w = {w}, where Z'(w) = {Z_deriv} and "
f"-2 * [1 + w * Z(w)] = {Z_deriv_characterization}.")
def test_plasma_dispersion_func(self, w, expected):
r"""Test plasma_dispersion_func against tabulated results and
known symmetry properties."""
# Many of the tabulated results originally came from the book
# entitled "The Plasma Dispersion Function: The Hilbert Transform
# of the Gaussian" by B. D. Fried and S. D. Conte (1961).
Z_of_w = plasma_dispersion_func(w)
assert np.isclose(
Z_of_w, expected, atol=1e-12 * (1 + 1j), rtol=1e-12
), (f"plasma_dispersion_func({w}) equals {Z_of_w} instead of the "
f"expected approximate result of {expected}. The difference between "
f"the actual and expected results is {Z_of_w - expected}.")
def test_normal_vs_lite(self, w, expected):
r"""Test that plasma_dispersion_func and plasma_dispersion_func_lite
calculate the same values."""
# Many of the tabulated results originally came from the book
# entitled "The Plasma Dispersion Function: The Hilbert Transform
# of the Gaussian" by B. D. Fried and S. D. Conte (1961).
Z_of_w = plasma_dispersion_func(w)
Z_of_w_lite = plasma_dispersion_func_lite(w)
assert np.isclose(
Z_of_w, Z_of_w_lite, atol=1e-12 * (1 + 1j), rtol=1e-12
), (f"plasma_dispersion_func({w}) and plasma_dispersion_func_lite({w}) "
"are not equal.")
def test_plasma_dispersion_func_power_series_expansion(self):
"""Test plasma_dispersion_func against a power series expansion of
the plasma dispersion function."""
w_array = np.array(
[
[0.1356 + 0.114j, -0.204 - 0.0012j],
[-0.131 + 0.131j, 0.1313 - 0.125j],
[-0.334 - 0.712j, 0.12411 + 0j],
[0.1278 + 0.928j, 0 + 0j],
],
dtype=np.complex128,
)
try:
Z_of_w_array = plasma_dispersion_func(w_array)
except Exception as exc:
raise ValueError(
"plasma_dispersion_func is unable to accept an "
f"ndarray argument with values:\n{w_array}") from exc
# The following power series expansion is given by equation (B.3)
# on page 401 of Plasma Waves by D. G. Swanson (2003, 2nd
# edition). The range of convergence of this expansion is not
# stated, but arguments are chosen to be close to the origin.
Z_power_series = np.zeros_like(w_array)
for n in range(0, 200):
Z_power_series += 1j * np.sqrt(π) * (1j * w_array)**n / Γ(n / 2 +
1)
assert np.allclose(
Z_of_w_array, Z_power_series, atol=1e-15 * (1 + 1j), rtol=1e-15
), ("The values returned by plasma_dispersion_func are inconsistent "
"with the power series expansion of the plasma dispersion function. "
f"The argument given to plasma_dispersion_func is:\n\n{w_array}\n\n"
f"The results of plasma_dispersion_func are:\n\n{Z_of_w_array}\n\n"
"The results from the power series expansion are:\n\n"
f"{Z_power_series}\n\n"
"The difference between these two results is:\n\n"
f"{Z_of_w_array - Z_power_series}\n")
def test_plasma_dispersion_func_deriv(w, expected):
r"""Test plasma_dispersion_func_deriv against tabulated results
and an exact relationship."""
# The tabulated results are taken from Fried & Conte (1961). The
# exact analytical relationship comes from the bottom of page 3 of
# Fried & Conte (1961).
Z_deriv = plasma_dispersion_func_deriv(w)
assert np.isclose(Z_deriv, expected, atol=5e-5 * (1 + 1j), rtol=5e-6), (
f"The derivative of the plasma dispersion function does not match "
f"the expected value for w = {w}. The value of "
f"plasma_dispersion_func_deriv({w}) equals {Z_deriv} whereas the "
f"expected value is {expected}. The difference between the actual "
f"and expected results is {Z_deriv - expected}.")
Z = plasma_dispersion_func(w)
Z_deriv_characterization = -2 * (1 + w * Z)
assert np.isclose(Z_deriv, Z_deriv_characterization, rtol=1e-15), (
f"The relationship that Z'(w) = -2 * [1 + w * Z(w)] is not "
f"met for w = {w}, where Z'(w) = {Z_deriv} and "
f"-2 * [1 + w * Z(w)] = {Z_deriv_characterization}.")
# w, expected_error
plasma_disp_func_errors_table = [
("", TypeError),
(7 * u.m, u.UnitsError),
(np.inf, ValueError),
(np.nan, ValueError),
]
# Array of expected values for plasma_dispersion_func
# (w, expected)
plasma_dispersion_func_table = [
(0, 1j * np.sqrt(π)),
(1, -1.076_159_013_825_536_8 + 0.652_049_332_173_292_2j),
(1j, 0.757_872_156_141_311_87j),
(1.2 + 4.4j, -0.054_246_157_069_223_27 + 0.207_960_584_359_855_62j),
(9.2j, plasma_dispersion_func(9.2j * u.dimensionless_unscaled)),
(5.4 - 3.1j, -0.139_224_873_051_713_11 - 0.082_067_822_640_155_802j),
(9.9 - 10j, 2.013_835_257_947_027_6 - 25.901_274_737_989_727j),
(4.5 - 10j, -1.367_495_046_340_094_7e35 - 6.853_923_234_842_270_6e34j),
]
class TestPlasmaDispersionFunction:
"""
Test class for `plasmapy.dispersion.plasma_dispersion_func`.
Note: Testing of `plasma_dispersion_func_lite` is done in a separate
test class.
"""
@pytest.mark.parametrize(
"bound_name, bound_attr",