Context: The Riemann-Hurwitz formula is a fundamental result in algebraic geometry and the theory of Riemann surfaces. It relates the genera of two Riemann surfaces connected by a holomorphic map, taking into account the ramification of the map. Problem Statement: Let f CD be a non-constant holomorphic (branched covering) map between compact, connected Riemann surfaces C and D. Suppose f has degree d, and let go and gD denote the genera of C and D, respectively. Tasks: 1. Definitions and Preliminary Concepts: a. Genus of a Riemann Surface: Define the genus g of a compact Riemann surface. Explain its topological significance in terms of the surface's connectivity and "holes." b. Holomorphic Maps and Degree: Define what it means for a map f : C→ D between Riemann surfaces to be holomorphic. Explain the concept of the degree d of the map f. c. Ramification Points and Indices: Define ramification points of f and ramification indices. Explain the ramification divisor associated with f.

Find three different bases for the 3-dimensional real space (ususally denoted R3).

Please help and show your work. thank you.

Question 3. Let V be a finite-dimensional complex vector space, and let f: VV be a linear map such that dim img f = 1. Prove that f is diagonalizable if and only if img fnker f = 0.

one example of a 4-dimensional vector space that is not R 4

Find a mapping G that maps the disk u² + ² ≤ 1 onto the interior of the ellipse ()² + (² ≤ 1. Identify the correct mapping. G(u, v) = (bu, av) G(u, v) = (au, bu) G(u, v) = (alu, b/v) G(u, v) = (ula, u/b) Then use the Change of Variables Formula to find the area A of the ellipse. (Express numbers in exact form. Use symbolic notation and fractions where needed.) A =

5). Let V be the space of meromorphic functions on the extended complex plane Ĉ that possibly have poles at 0, 1 and ∞ with orders up to 2. Find a basis for V.

need help with complex analysis please, thankyou

Could you please help me out with question 8?

1. Norms and inner products – Cauchy-Schwarz and triangle inequalities, Pythagorean theorem, parallelogram equality: (a) Use the general properties of an inner product (·, ·) on a vector space V, with induced norm ||u|| = V(u, u), to show the Cauchy-Schwarz inequality: For any u, v E V, |(u, v)| 0 (justify this). Expand this inner prod- uct to get an expression that is quadratic in t; show that the condition for this quadratic to remain non-negative for all t is equivalent to (CS). Of course for the geometric dot product, involving the cosine of an angle, (CS) can be shown very easily...] (b) For any norm || · || induced by an inner product, as in (a) above, prove the triangle inequality: For any u, v E V, ||u+ v|| < ||u|| + ||v||, (A) with equality iff u and v are orthogonal. [Hint: Square both sides. Of course for || · | to be a norm it must satisfy (A); this result shows that for any inner product, the quantity V(u, u) indeed defines a norm.]