Q1. Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.
- i)) If z = a + ib, where a and b are integers, then |1+ z + z² +…+ zⁿ | ≥ | z |ⁿ if a > 0. (80 words)
- ii)) If f(z) and f(z) are analytic functions in a domain, then f is necessarily a constant. (80 words)
- iii)) A real-valued function u(x, y) is harmonic in D if u(x, – y) is harmonic in D. (80 words)
- iv)) lim (n!)¹/ⁿ = ∞. (as n approaches infinity) (80 words)
- v)) The inequality | eᵃ -eᵇ |≤ | a-b| holds for a, b ∈ D = {w : Re w ≤ 0}. (80 words)
- vi)) If f(z) = Sum from n=0 to infinity of a_n (z-a)ⁿ has the property that Sum from n=0 to infinity of f⁽ⁿ⁾(a) converges, then f is necessarily an entire function. (80 words)
- vii)) If a power series Sum from n=0 to infinity of a_n zⁿ converges for |z|<1 and if b_n ∈ C is such that | b_n | < n² | a_n | for all n≥0, then Sum from n=0 to infinity of b_n zⁿ converges for |z|<1. (80 words)
- viii)) If f is entire and f(z) = f(−z) for all z, then there exists an entire function g such that f(z) = g(z²) for all z ∈ C. (80 words)
- ix)) A mobius transformation which maps the upper half plane {z: Im z > 0} onto itself and fixing 0,∞ and no other points, must be of the form T_z = az for some a not equal to 0 and alpha not equal to 1. (80 words)
- x)) If f is entire and Re f(z) is bounded as | z | approaches infinity, then f is constant. (80 words)
Key Points:
- Analytic f(z) and f(z) implies f is constant via CR equations.
- Reflection u(x,-y) preserves harmonicity of u(x,y).
- Limit (n!)^(1/n) as n→∞ is infinity by Stirling's Approximation.
- Inequality |e^a - e^b| ≤ |a-b| holds for Re(a), Re(b) ≤ 0.
Answer: This response comprehensively addresses ten sub-questions related to complex analysis, determining the truth value of each statement and providing concise justifications with proofs or counter-examples. The solutions leverage fundamental theorems and concepts from the MMT-005 Complex Analysis course, including Cauchy-Riemann equations, properties of harmonic functions, Stirling's approximation, Mean Value Inequality for complex integrals, radius of convergence of power series, Liouville's Theore...