Discussion
Let \(f(z) = \sum_{n=1}^{\infty} 5^{-n} \cos(nz)\). On which one of the following domains, does \(f(z)\) represent an analytic function?
Answer: A
f is analytic on \(\{z \in \mathbb{C} : |\text{Im}\, z| \lt \ln 5\}\).
For \(z = x + iy\), \(|\cos(nz)|\) grows like \(\frac{1}{2} e^{n|y|}\). The terms therefore behave like
\[ 5^{-n} e^{n|y|} = \left( \frac{e^{|y|}}{5} \right)^{n} \]
a geometric series that converges when \(e^{|y|} \lt 5\), that is \(|y| \lt \ln 5\), and diverges when \(|y| \gt \ln 5\). On any strip \(|\text{Im}\, z| \le r \lt \ln 5\) the convergence is uniform (Weierstrass M-test), and a uniform limit of analytic functions is analytic.
Outside the strip the series diverges, and conditions on \(\text{Re}\, z\) are irrelevant because \(\cos(nz)\) is bounded in the real direction but not in the imaginary one.
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