Complex Analysis (Practice Exam)

In summary, the problem is to compute the 100th derivative of the function f(z) = 1/(1+i-sqrt(2)z) evaluated at 0, using Cauchy's integral formula. The answer is -1/(1+i). To solve this, the function can be written as a series representation about 0, and the desired calculation is the coefficient in front of the z^100 term.
  • #1
PieceOfPi
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Homework Statement



This question is in my exam review problem from my complex analysis class.
Compute f(100)(0)/100!, where f(z) = 1/(1+i-sqrt(2)z).

(f(100)(0) means the 100th derivative of f evaluated at 0.)

Homework Equations



Cauchy's integral formula might be helpful.

The answer to this question is -1/(1+i).

The Attempt at a Solution



I thought about using Cauchy's integral formula, but it seems like I need to rewrite f(z) so that I have singularity at 0... or maybe I'm completely missing out.

Thanks!
 
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  • #2
Have you tried writing the function as it's series representation about 0? Then the calculation you are looking for would be the coefficient in front of the z^100 term.
 

1. What is the purpose of complex analysis?

Complex analysis is a branch of mathematics that studies the properties and behavior of complex numbers and functions. It is used to solve problems in various fields, including physics, engineering, and economics.

2. What are complex numbers?

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

3. What are the fundamental theorems of complex analysis?

The fundamental theorems of complex analysis include the Cauchy-Riemann equations, Cauchy's integral theorem, and Cauchy's integral formula.

4. How is complex analysis used in engineering?

Complex analysis is used in engineering to analyze and design systems with complex variables, such as electronic circuits and control systems. It is also used in fluid mechanics and heat transfer problems.

5. What are some real-world applications of complex analysis?

Complex analysis has many real-world applications, including analyzing fluid flow, designing electrical circuits, and solving problems in quantum mechanics. It is also used in signal processing, image processing, and finance.

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