1. Coolant Level: Make sure the coolant level in the expansion tank is between the "MIN" and "MAX" marks. If it's low, add a mixture of 50% coolant and 50% distilled water.
2. Radiator Fan: Check if the radiator fan is working properly. Turn on the engine and let it reach operating temperature. The fan should turn on automatically to cool the radiator. If it doesn't, there may be an electrical issue or a faulty fan motor.
3. Thermostat: The thermostat regulates the flow of coolant to the radiator. A faulty thermostat can prevent the coolant from circulating properly, causing overheating. Check the thermostat by removing it and placing it in a pot of boiling water. If it doesn't open when the water reaches boiling point, it needs to be replaced.
4. Water Pump: The water pump circulates the coolant throughout the engine. A faulty water pump can reduce the flow of coolant, leading to overheating. Check the water pump for leaks or damage. If necessary, replace the water pump and its gasket.
5. Radiator: The radiator is responsible for cooling the coolant. Check the radiator for leaks, clogs, or damage. If the radiator is blocked, it can prevent the coolant from flowing properly and cause overheating. Clean or replace the radiator as needed.
If you've checked all these things and your BMW e39 is still overheating, you should take it to a qualified mechanic for further diagnosis and repair.