数学期末考试练习代写 Math 132代写 数学考试助攻 数学代写

Math 132 Final exam practice

数学期末考试练习代写 Directions. This is not an assignment to be turned in. These questions are meant to provide practice for the final exam.

Directions. This is not an assignment to be turned in. These questions are meant to provide practice for the final exam.

Note that these problems are not meant to give a comprehensive review of the material. You should still use old homeworks, old midterms and practice midterms, and textbook problems to also help you prepare.

Problem 2.

(a) Let f(z) be a function and let z₀ ∈ C. Define what it means for f(z) to be analytic at z₀.

(b) Define the function

f(z) = f(x + iy) = (x³ + 3xy² − 6y² + x) + i(y³ + 3x²y + y).

Find all the points z ∈ C at which f is analytic.

Problem 3. 数学期末考试练习代写

Consider a real-valued function u(x, y), where x and y are real variables. For each way of defining u(x, y) below, determine whether there exists a real-valued function v(x, y) such that f(z) = u(x, y) + iv(x, y) is a function analytic in some domain D ⊆ C. If such a v(x, y) exists, find one such and determine the domain of analyticity D for f(z). If such a v(x, y) does not exist, prove that it does not exist.

(a) u(x, y) = y³ − 3x²y

(b) u(x, y) = xy² − x²y

(c) u(x, y) = Arg(x + iy).

Hint. For (c), it will be helpful to begin by considering the definition of Log (z).

Problem 6. 数学期末考试练习代写

Suppose f(z) is an entire function and that there exists a real number R such that

|f(z)| ≤ |z| for any complex number z with |z| ≥ R

Prove that f is of the form f(z) = a + bz.

Hint. Use an ML estimate argument to show lots of the coefficients in the power series expansion of f(z) are equal to 0.

Problem 7. 数学期末考试练习代写

Determine whether statement (∗) below is true or false. If the statement is true, then prove it; if the statement is false, provide a counterexample that shows it is false.

(∗) Let h(z) be a function which has an isolated singularity at z₀. If h(z) has an essential singularity at z₀, then h’(z) has an essential singularity at z₀.

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