Math Problem Statement

1. The set |z + 2 + 5i| ≤ 1/2 in the complex plane represents a closed circular disk with center and radius ... 2. A function f(z) is said to be analytic in a domain D if f(z) is ... and ... at all points of D. 3. Cauchy-Riemann equations in the Cartesian coordinate system are ... and ...

Solution

Let's analyze each of these multiple-choice questions one by one.

1. First Question

  • Problem: The set z+2+5i12|z + 2 + 5i| \leq \frac{1}{2} in the complex plane represents a closed circular disk with a center and radius.
  • Solution:
    • The given set z+2+5i12|z + 2 + 5i| \leq \frac{1}{2} describes a disk in the complex plane.
    • We can rewrite this as z(25i)12|z - (-2 - 5i)| \leq \frac{1}{2}.
    • This means the center of the disk is (2,5)(-2, -5), and the radius is 12\frac{1}{2}.
  • Correct Answer: (d) (2,5),12(-2, -5), \frac{1}{2}.

2. Second Question

  • Problem: A function f(z)f(z) is said to be analytic in a domain DD if f(z)f(z) is ... and ... at all points of DD.
  • Solution:
    • A function is analytic in a domain DD if it is both defined and differentiable at all points in DD.
  • Correct Answer: (b) defined, differentiable.

3. Third Question

  • Problem: Cauchy-Riemann equations in the Cartesian coordinate system are ... and ...
  • Solution:
    • The Cauchy-Riemann equations are given by ux=vyu_x = v_y and uy=vxu_y = -v_x, where uu and vv are the real and imaginary parts of a complex function f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y).
  • Correct Answer: (c) ux=vy,uy=vxu_x = v_y, u_y = -v_x.

Would you like further explanations on any of these answers, or is there anything specific you'd like me to elaborate on?


Here are five related questions that might help expand your understanding:

  1. How do you determine the center and radius of a disk represented by a complex inequality?
  2. What are the necessary and sufficient conditions for a function to be analytic in a domain?
  3. Can you explain the geometric interpretation of the Cauchy-Riemann equations?
  4. Why is differentiability in the complex sense a stronger condition than in the real sense?
  5. How does the concept of analyticity relate to complex differentiability and continuity?

Tip: In complex analysis, always verify the differentiability of a function using the Cauchy-Riemann equations for insight into its analytic nature.

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Math Problem Analysis

Mathematical Concepts

Complex Analysis
Analytic Functions
Cauchy-Riemann Equations
Closed Disks in Complex Plane

Formulas

|z - z₀| ≤ r for disks in the complex plane
Conditions for analyticity: function must be defined and differentiable
Cauchy-Riemann Equations: u_x = v_y and u_y = -v_x

Theorems

Definition of Analyticity in Complex Analysis
Cauchy-Riemann Equations

Suitable Grade Level

Undergraduate - Complex Analysis