Discussion
Consider the following statements :
(a) Every non-negative harmonic function on \(\mathbf{C}\) is constant.
(b) Every harmonic function on \(\mathbf{C} - \{0\}\) has a harmonic conjugate
Then :
Answer: A
The answer is only (a) is true.
(a) is Liouville's theorem for harmonic functions: a harmonic function on the whole plane that is bounded below (here by 0) is constant. One proof applies Harnack's inequality, or writes \(u = \mathrm{Re} f\) with \(f\) entire and applies Liouville to \(e^{-f}\), which is bounded.
(b) is false because \(\mathbf{C} - \{0\}\) is not simply connected. The function \(\log|z|\) is harmonic there, but its harmonic conjugate would have to be \(\arg z\), which cannot be defined continuously around the origin.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development