Complex Analysis

Step 1: Understanding the Concept:
This question requires the evaluation of four different complex contour integrals using various methods of complex analysis, including Cauchy's Integral Formula, parameterization, and the Residue Theorem.

Step 2: Detailed Explanation:
A. The integral is over a simple closed contour enclosing the point . Let . By Cauchy's Integral Formula, . Therefore, A matches III.
B. The contour is a semi-circle from to , so it's not a closed path. We must evaluate by parameterization. Let , so . The limits are . Therefore, B matches I.
C. The contour is , which is a circle that encloses both simple poles at and . We use the Residue Theorem. Let . By the Residue Theorem, the integral is . Therefore, C matches II.
D. The contour is , which encloses the simple pole at . We use Cauchy's Integral Formula. Let . The integral is . Therefore, D matches IV.

Step 3: Final Answer:
The correct matching is A-III, B-I, C-II, D-IV. This corresponds to option (C).

Step 1: Understanding the Concept:
This question tests key theorems and properties concerning the zeros of analytic functions in complex analysis.

Step 2: Detailed Explanation:
A. The order of a zero of a polynomial equals to the order of its first non-vanishing derivative at that zero of the polynomial.
This is correct. If an analytic function has a zero of order at , its Taylor series expansion around starts with the term , where . This implies that and . So, the order of the zero is indeed the order of the first non-vanishing derivative. Statement A is correct.
B. Zeros of non-zero analytic function are isolated.
This is a fundamental property of analytic functions, often called the Identity Theorem or Uniqueness Principle. It states that for any zero of a non-zero analytic function , there exists a punctured disk centered at where is non-zero. Statement B is correct.
C. Zeros of f(z) are obtained by equating the numerator to zero if there is no common factor in the numerator and the denominator of f(z).
This statement applies to a rational function . A zero of occurs at a point where . This requires and . If there were a common factor , it would be a removable singularity, not a zero. Statement C is correct.
D. Limit points of zeros of an analytic function is an isolated essential singularity.
This statement is incorrect. By the Identity Theorem, if the set of zeros of an analytic function has a limit point within the domain of analyticity, then must be identically zero on the entire domain. Therefore, a non-zero analytic function cannot have a limit point of zeros in its domain. Statement D is incorrect.

Step 3: Final Answer:
Statements A, B, and C are correct. Therefore, the correct option is (B).

Step 1: Understanding the Concept:
A complex function is differentiable at a point if it satisfies the Cauchy-Riemann equations at that point and its first-order partial derivatives are continuous there. Let , so . The Cauchy-Riemann equations are and .

Step 2: Key Formula or Approach:
First, express the function in terms of its real and imaginary parts. So, the real part is and the imaginary part is .

Step 3: Detailed Explanation:
Now, we compute the first-order partial derivatives: For the Cauchy-Riemann equations to hold, we need: 1.
2.
Both equations are satisfied only when and . This corresponds to the point .
The partial derivatives are polynomials in and , so they are continuous everywhere. Since the Cauchy-Riemann equations are satisfied only at and the partials are continuous, the function is differentiable only at .

Step 4: Final Answer:
The function is differentiable only at .

Step 1: Understanding the Concept:
This problem involves evaluating a complex contour integral. The key is to analyze the integrand and the position of its singularities relative to the integration contour.

Step 2: Key Formula or Approach:
The problem is solved using Cauchy's Integral Theorem. The theorem states that if a function is analytic at all points inside and on a simple closed contour C, then the integral of over C is zero.
Step 3: Detailed Explanation:
The integrand is the function . The numerator, , is a polynomial and thus is analytic everywhere in the complex plane (it is an entire function). The denominator, , is zero only at . Therefore, the integrand has a single singularity (a pole of order 3) at the point . The problem statement specifies that the point is outside the closed curve C. This means that the function is analytic at all points inside and on the contour C. Therefore, by Cauchy's Integral Theorem, the integral must be zero.

Step 4: Final Answer:
The value of the integral is 0.

Step 1: Understanding the Concept:
This problem requires evaluating a complex line integral along a specified path. The standard method is to parameterize the path and convert the complex integral into an integral of a single real variable.

Step 2: Key Formula or Approach:
1. Parameterize the path. The path is . We can use as the parameter. The path starts at (i.e., ) and ends at (i.e., ). So, goes from 0 to 1. 2. Express everything in terms of the parameter . - - - The integrand is 3. Substitute into the integral and evaluate.
Step 3: Detailed Explanation:
We evaluate the integral by first expanding the product inside: Now integrate term by term: Now, multiply the complex numbers:
Step 4: Final Answer:
The value of the integral is . This corresponds to option (B). (Note: If the path were , the answer would be , which is option A. It's possible the question had a typo.)

Step 1: Understanding the Concept:
An analytic function has a real part and imaginary part that are harmonic conjugates, meaning they satisfy the Cauchy-Riemann equations: and . Given one part (e.g., ), we can find the other part ( ) and construct the analytic function using the Milne-Thomson method.

Step 2: Key Formula or Approach (Milne-Thomson Method):
If the real part is given, the derivative of the analytic function can be expressed as: In the Milne-Thomson method, we replace with and with in this expression to get directly in terms of . Then, we integrate to find .

Step 3: Detailed Explanation:
Given . First, find the partial derivatives: Now, use the Milne-Thomson method. Replace with and with : Substitute these into the formula for : Now, integrate with respect to to find : Let's see if this matches any options. It matches the form of option (D). Let's check the result. If , let . The real part . This matches the given (up to a constant which can be absorbed into C). The solution can be written as where is another arbitrary complex constant.

Step 4: Final Answer:
The corresponding analytic function is .

Step 1: Understanding the Concept:
This problem involves an analytic function given in terms of and . We need to find its derivative and express it in terms of to identify the constant . A good strategy is to first express in terms of and , and then check for analyticity. An easier way is to express directly in terms of by recognizing combinations of and .

Step 2: Express f(z) in terms of z:
Let and . We know that . Let's try to construct from . Consider the complex number . This matches the given . So, .

Step 3: Find the Derivative and the Constant c:
Since is a polynomial in , it is analytic everywhere. We can differentiate it directly with respect to : We are given that . By comparing the two expressions for , we find that the constant is:
Step 4: Calculate the Magnitude |c|:
The magnitude (or modulus) of a complex number is .