Then you do not need a residence permit for studying in Germany.

If your visa does not cover your whole stay in Germany, you must apply for a residence permit at the local immigration office at least two months before your visa expires.

These are the documents you have to hand in, when applying for a residence permit:

• Valid passport with visa and photocopies of all printed pages
• Up-to-date digital biometrical photograph
• Proof of health insurance coverage
• Proof of enrolment at the University of Cologne
• Proof of financial resources to cover your stay in Germany
• Copy of rental contract
• Certificate of residence registration ("Meldebescheinigung")

You are citizen of Australia, Canada, Israel, Japan, New Zealand, the Republic of Korea or the United States of America and you entered Germany WITHOUT a visa:
Then you must apply for a residence permit at the local immigration office within 90 days after entering Germany.