(ph0SSKJr SSKJr SSKrSSKJrJrJ r J r J r J r J r Jr SSjr\"S5SSj5r\"S5SS j5r\"S5SS j5rSS jr\"\S 9SS j5rg)wraps) _lazywhereN) _spherical_jn _spherical_yn _spherical_in _spherical_kn_spherical_jn_d_spherical_yn_d_spherical_in_d_spherical_kn_dc^^UU4SjnU$)NcJ>^^UU4Sjm[T5SUUU4Sjj5nU$)Nct>[R"US-S:HTT*5nU(aU*OUnT"X*U5U-$)Nr)npwhere)nz derivativesignfun sign_n_evens R/var/www/html/venv/lib/python3.13/site-packages/scipy/special/_spherical_bessel.pystandard_reflection>use_reflection..decorator..standard_reflections>88AEQJ k\BD&D5DDq"j)D0 0c >^^[R"U5n[R"UR[R5(a T"XT5$TcTOTm[ UR S:X4UU4SjUU4SjS9S$)Nrc>T"XT5$N)rrrrs rDuse_reflection..decorator..wrapper.. s Sz-Brc>T"XT5$r!r")rrrf2s rr#r$!s bz.Br)fr&r")rasarray issubdtypedtypecomplexfloatingrreal)rrrr&rreflection_funrs `@rwrapper2use_reflection..decorator..wrappersq 1 A}}QWWb&8&8991,,(6(>$NBaffkA6 B!BDDFH HrFr)rr.rr-rs` @r decorator!use_reflection..decorators/ 1 s H H  Hrr")rr-r1s`` ruse_reflectionr3 s * rc[R"U[R"S5S9nU(a [X5$[ X5$)aSpherical Bessel function of the first kind or its derivative. Defined as [1]_, .. math:: j_n(z) = \sqrt{\frac{\pi}{2z}} J_{n + 1/2}(z), where :math:`J_n` is the Bessel function of the first kind. Parameters ---------- n : int, array_like Order of the Bessel function (n >= 0). z : complex or float, array_like Argument of the Bessel function. derivative : bool, optional If True, the value of the derivative (rather than the function itself) is returned. Returns ------- jn : ndarray Notes ----- For real arguments greater than the order, the function is computed using the ascending recurrence [2]_. For small real or complex arguments, the definitional relation to the cylindrical Bessel function of the first kind is used. The derivative is computed using the relations [3]_, .. math:: j_n'(z) = j_{n-1}(z) - \frac{n + 1}{z} j_n(z). j_0'(z) = -j_1(z) .. versionadded:: 0.18.0 References ---------- .. [1] https://dlmf.nist.gov/10.47.E3 .. [2] https://dlmf.nist.gov/10.51.E1 .. [3] https://dlmf.nist.gov/10.51.E2 .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- The spherical Bessel functions of the first kind :math:`j_n` accept both real and complex second argument. They can return a complex type: >>> from scipy.special import spherical_jn >>> spherical_jn(0, 3+5j) (-9.878987731663194-8.021894345786002j) >>> type(spherical_jn(0, 3+5j)) We can verify the relation for the derivative from the Notes for :math:`n=3` in the interval :math:`[1, 2]`: >>> import numpy as np >>> x = np.arange(1.0, 2.0, 0.01) >>> np.allclose(spherical_jn(3, x, True), ... spherical_jn(2, x) - 4/x * spherical_jn(3, x)) True The first few :math:`j_n` with real argument: >>> import matplotlib.pyplot as plt >>> x = np.arange(0.0, 10.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-0.5, 1.5) >>> ax.set_title(r'Spherical Bessel functions $j_n$') >>> for n in np.arange(0, 4): ... ax.plot(x, spherical_jn(n, x), label=rf'$j_{n}$') >>> plt.legend(loc='best') >>> plt.show() longr*)rr(r*r rrrrs r spherical_jnr8&s7f 1BHHV,-Aq$$Q""rc[R"U[R"S5S9nU(a [X5$[ X5$)aWSpherical Bessel function of the second kind or its derivative. Defined as [1]_, .. math:: y_n(z) = \sqrt{\frac{\pi}{2z}} Y_{n + 1/2}(z), where :math:`Y_n` is the Bessel function of the second kind. Parameters ---------- n : int, array_like Order of the Bessel function (n >= 0). z : complex or float, array_like Argument of the Bessel function. derivative : bool, optional If True, the value of the derivative (rather than the function itself) is returned. Returns ------- yn : ndarray Notes ----- For real arguments, the function is computed using the ascending recurrence [2]_. For complex arguments, the definitional relation to the cylindrical Bessel function of the second kind is used. The derivative is computed using the relations [3]_, .. math:: y_n' = y_{n-1} - \frac{n + 1}{z} y_n. y_0' = -y_1 .. versionadded:: 0.18.0 References ---------- .. [1] https://dlmf.nist.gov/10.47.E4 .. [2] https://dlmf.nist.gov/10.51.E1 .. [3] https://dlmf.nist.gov/10.51.E2 .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- The spherical Bessel functions of the second kind :math:`y_n` accept both real and complex second argument. They can return a complex type: >>> from scipy.special import spherical_yn >>> spherical_yn(0, 3+5j) (8.022343088587197-9.880052589376795j) >>> type(spherical_yn(0, 3+5j)) We can verify the relation for the derivative from the Notes for :math:`n=3` in the interval :math:`[1, 2]`: >>> import numpy as np >>> x = np.arange(1.0, 2.0, 0.01) >>> np.allclose(spherical_yn(3, x, True), ... spherical_yn(2, x) - 4/x * spherical_yn(3, x)) True The first few :math:`y_n` with real argument: >>> import matplotlib.pyplot as plt >>> x = np.arange(0.0, 10.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-2.0, 1.0) >>> ax.set_title(r'Spherical Bessel functions $y_n$') >>> for n in np.arange(0, 4): ... ax.plot(x, spherical_yn(n, x), label=rf'$y_{n}$') >>> plt.legend(loc='best') >>> plt.show() r5r6)rr(r*r rr7s r spherical_ynr;s7d 1BHHV,-Aq$$Q""rc[R"U[R"S5S9nU(a [X5$[ X5$)aModified spherical Bessel function of the first kind or its derivative. Defined as [1]_, .. math:: i_n(z) = \sqrt{\frac{\pi}{2z}} I_{n + 1/2}(z), where :math:`I_n` is the modified Bessel function of the first kind. Parameters ---------- n : int, array_like Order of the Bessel function (n >= 0). z : complex or float, array_like Argument of the Bessel function. derivative : bool, optional If True, the value of the derivative (rather than the function itself) is returned. Returns ------- in : ndarray Notes ----- The function is computed using its definitional relation to the modified cylindrical Bessel function of the first kind. The derivative is computed using the relations [2]_, .. math:: i_n' = i_{n-1} - \frac{n + 1}{z} i_n. i_1' = i_0 .. versionadded:: 0.18.0 References ---------- .. [1] https://dlmf.nist.gov/10.47.E7 .. [2] https://dlmf.nist.gov/10.51.E5 .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- The modified spherical Bessel functions of the first kind :math:`i_n` accept both real and complex second argument. They can return a complex type: >>> from scipy.special import spherical_in >>> spherical_in(0, 3+5j) (-1.1689867793369182-1.2697305267234222j) >>> type(spherical_in(0, 3+5j)) We can verify the relation for the derivative from the Notes for :math:`n=3` in the interval :math:`[1, 2]`: >>> import numpy as np >>> x = np.arange(1.0, 2.0, 0.01) >>> np.allclose(spherical_in(3, x, True), ... spherical_in(2, x) - 4/x * spherical_in(3, x)) True The first few :math:`i_n` with real argument: >>> import matplotlib.pyplot as plt >>> x = np.arange(0.0, 6.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-0.5, 5.0) >>> ax.set_title(r'Modified spherical Bessel functions $i_n$') >>> for n in np.arange(0, 4): ... ax.plot(x, spherical_in(n, x), label=rf'$i_{n}$') >>> plt.legend(loc='best') >>> plt.show() r5r6)rr(r*r r r7s r spherical_inr=7b 1BHHV,-Aq$$Q""rc0[XS-US9R$)Ny)r) spherical_knr,r7s rspherical_kn_reflectionrA1s r6j 9 > >>r)r-c[R"U[R"S5S9nU(a [X5$[ X5$)aModified spherical Bessel function of the second kind or its derivative. Defined as [1]_, .. math:: k_n(z) = \sqrt{\frac{\pi}{2z}} K_{n + 1/2}(z), where :math:`K_n` is the modified Bessel function of the second kind. Parameters ---------- n : int, array_like Order of the Bessel function (n >= 0). z : complex or float, array_like Argument of the Bessel function. derivative : bool, optional If True, the value of the derivative (rather than the function itself) is returned. Returns ------- kn : ndarray Notes ----- The function is computed using its definitional relation to the modified cylindrical Bessel function of the second kind. The derivative is computed using the relations [2]_, .. math:: k_n' = -k_{n-1} - \frac{n + 1}{z} k_n. k_0' = -k_1 .. versionadded:: 0.18.0 References ---------- .. [1] https://dlmf.nist.gov/10.47.E9 .. [2] https://dlmf.nist.gov/10.51.E5 .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- The modified spherical Bessel functions of the second kind :math:`k_n` accept both real and complex second argument. They can return a complex type: >>> from scipy.special import spherical_kn >>> spherical_kn(0, 3+5j) (0.012985785614001561+0.003354691603137546j) >>> type(spherical_kn(0, 3+5j)) We can verify the relation for the derivative from the Notes for :math:`n=3` in the interval :math:`[1, 2]`: >>> import numpy as np >>> x = np.arange(1.0, 2.0, 0.01) >>> np.allclose(spherical_kn(3, x, True), ... - 4/x * spherical_kn(3, x) - spherical_kn(2, x)) True The first few :math:`k_n` with real argument: >>> import matplotlib.pyplot as plt >>> x = np.arange(0.0, 4.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(0.0, 5.0) >>> ax.set_title(r'Modified spherical Bessel functions $k_n$') >>> for n in np.arange(0, 4): ... ax.plot(x, spherical_kn(n, x), label=rf'$k_{n}$') >>> plt.legend(loc='best') >>> plt.show() r5r6)rr(r*rr r7s rr@r@8r>r)NNr0) functoolsrscipy._lib._utilrnumpyr_ufuncsrrr r r r r rr3r8r;r=rAr@r"rrrGs'888 :V#V#rU#U#pT#T#n?67T#8T#r