def flow_para_function_with_vibration( x, beta, relaxation_rate, flow_velocity, freq, amp, baseline=1):
vibration_part = (1 + amp*np.cos( 2*np.pi*freq* x) )
Diff_part= np.exp(-2 * relaxation_rate * x)
Flow_part = np.pi**2/(16*x*flow_velocity) * abs( erf( np.sqrt( 4/np.pi * 1j* x * flow_velocity ) ) )**2
return beta* vibration_part* Diff_part * Flow_part + baseline
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