code: def test_besselk_series():
      const = log(2) - S.EulerGamma - log(x)
      assert besselk(0, x).series(x, n=4) == const + x**2 * (
          -log(x) / 4 - S.EulerGamma / 4 + log(2) / 4 + S(1) / 4
      ) + O(x**4 * log(x))
      assert besselk(1, x).series(x, n=4) == 1 / x + x * (
          log(x) / 2 - log(2) / 2 - S(1) / 4 + S.EulerGamma / 2
      ) + x**3 * (
          log(x) / 16 - S(5) / 64 - log(2) / 16 + S.EulerGamma / 16
      ) + O(
          x**4 * log(x)
      )
      assert besselk(2, x).series(x, n=4) == 2 / x**2 - S(
          1
      ) / 2 + x**2 * (
          -log(x) / 8 - S.EulerGamma / 8 + log(2) / 8 + S(3) / 32
      ) + O(
          x**4 * log(x)
      )
      assert besselk(0, x ** (1.1)).series(x, n=4) == log(
          2
      ) - S.EulerGamma - 1.1 * log(x) + x**2.2 * (
          -0.275 * log(x) - S.EulerGamma / 4 + log(2) / 4 + S(1) / 4
      ) + O(
          x**4 * log(x)
      )
      assert besselk(0, x**2 + x).series(x, n=4) == const + (
          2 - 2 * S.EulerGamma
      ) * (x**3 / 4 + x**2 / 8) - x + x**2 * (
          -log(x) / 4 + log(2) / 4 + S(1) / 2
      ) + x**3 * (
          -log(x) / 2 - S(7) / 12 + log(2) / 2
      ) + O(
          x**4 * log(x)
      )
      assert besselk(0, x / (1 - x)).series(
          x, n=3
      ) == const - x + x**2 * (
          -log(x) / 4 - S(1) / 4 - S.EulerGamma / 4 + log(2) / 4
      ) + O(
          x**3 * log(x)
      )
      assert besselk(0, log(1 + x)).series(
          x, n=3
      ) == const + x / 2 + x**2 * (
          -log(x) / 4 - S.EulerGamma / 4 + S(1) / 24 + log(2) / 4
      ) + O(
          x**3 * log(x)
      )
      assert besselk(1, 2 * sqrt(x)).series(x, n=3) == 1 / (
          2 * sqrt(x)
      ) + sqrt(x) * (log(x) / 2 - S(1) / 2 + S.EulerGamma) + x ** (
          S(3) / 2
      ) * (
          log(x) / 4 - S(5) / 8 + S.EulerGamma / 2
      ) + x ** (
          S(5) / 2
      ) * (
          log(x) / 24 - S(5) / 36 + S.EulerGamma / 12
      ) + O(
          x**3 * log(x)
      )
      assert besselk(-2, sin(x)).series(x, n=4) == besselk(
          2, sin(x)
      ).series(x, n=4)
  
imports: from itertools import product
  from sympy.concrete.summations import Sum
  from sympy.core.function import diff, expand_func
  from sympy.core.numbers import I, Rational, oo, pi
  from sympy.core.singleton import S
  from sympy.core.symbol import Symbol, symbols
  from sympy.functions.elementary.complexes import conjugate, polar_lift
  from sympy.functions.elementary.exponential import exp, exp_polar, log
  from sympy.functions.elementary.hyperbolic import cosh, sinh
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.functions.elementary.trigonometric import cos, sin
  from sympy.functions.special.bessel import besseli, besselj, besselk, bessely, hankel1, hankel2, hn1, hn2, jn, jn_zeros, yn
  from sympy.functions.special.gamma_functions import gamma, uppergamma
  from sympy.functions.special.hyper import hyper
  from sympy.integrals.integrals import Integral
  from sympy.series.order import O
  from sympy.series.series import series
  from sympy.functions.special.bessel import airyai, airybi, airyaiprime, airybiprime, marcumq
  from sympy.core.random import random_complex_number, verify_numerically, test_derivative_numerically, _randint
  from sympy.simplify import besselsimp
  from sympy.testing.pytest import raises, slow
  from sympy.abc import z, n, k, x
label_desc: Use image node_id 10 for calling a global function with example usage: test_besselk_series() without return types