code: def test_constructor_with_without_freq(self):
      # GH53687
      start = Period("2002-01-01 00:00", freq="30min")
      exp = period_range(start=start, periods=5, freq=start.freq)
      result = period_range(start=start, periods=5)
      tm.assert_index_equal(exp, result)
  
imports: import numpy
  import pytest
  from pandas._libs.tslibs.period import IncompatibleFrequency
  from pandas.core.dtypes.dtypes import PeriodDtype
  from pandas import Index, NaT, Period, PeriodIndex, Series, date_range, offsets, period_range
  import pandas._testing
  from pandas.core.arrays import PeriodArray
label_desc: Use image node_id 12 for calling the TestPeriodIndex obj's underlying member method code with example usage: obj.test_constructor_with_without_freq() without return types