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# Cauchy-Goursat Theorem

Not what you're looking for?

Given: Integral from zero to infinity of cos(x^2) dx = integral from zero to infinity of sin(x^2) dx = 1/2 * (square root of pi/2).

These can be evaluated by considering cos(x^2) = Re(e^ix^2) and sin(x^2) = Im(e^ix^2).

1.) Integrate the function f(z) = e^i(z)^2 around the positively oriented boundary of the sector 0 <= r <= R, 0 <= theta <= pi/4, and using the Cauchy-Goursat Theorem show that:

the integral from 0 to R: cos(x^2)dx =

1 / (2)^1/2 * [integral from 0 to R: e^-r^2 dr] - Re * [integral on CR: e^i(z)^2 dz]

2.) Show that the value of the integral along the arc CR tends to zero as R tends to infinity by obtaining the inequality:

| integral on CR of e^i(z^2) dz | <= (R/2) * [integral from 0 to (pi/2) of e^(-R^2) * sin(theta) d(theta).

##### Solution Summary

Cauchy-Goursat Theorem is utilized.

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
The text passage contains a mathematical problem involving the Cauchy-Goursat Theorem and the evaluation of specific integrals. It provides a problem statement and asks to prove certain results, but does not provide the answers to these questions.
multiple_questions
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Purchase Solution

# Cauchy-Goursat Theorem

Not what you're looking for?

Given: Integral from zero to infinity of cos(x^2) dx = integral from zero to infinity of sin(x^2) dx = 1/2 * (square root of pi/2).

These can be evaluated by considering cos(x^2) = Re(e^ix^2) and sin(x^2) = Im(e^ix^2).

1.) Integrate the function f(z) = e^i(z)^2 around the positively oriented boundary of the sector 0 <= r <= R, 0 <= theta <= pi/4, and using the Cauchy-Goursat Theorem show that:

the integral from 0 to R: cos(x^2)dx =

1 / (2)^1/2 * [integral from 0 to R: e^-r^2 dr] - Re * [integral on CR: e^i(z)^2 dz]

2.) Show that the value of the integral along the arc CR tends to zero as R tends to infinity by obtaining the inequality:

| integral on CR of e^i(z^2) dz | <= (R/2) * [integral from 0 to (pi/2) of e^(-R^2) * sin(theta) d(theta).

##### Solution Summary

Cauchy-Goursat Theorem is utilized.

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
user
The text passage contains a mathematical problem involving the Cauchy-Goursat Theorem and the evaluation of specific integrals. It provides a problem statement and asks to prove certain results, but does not provide the answers to these questions.

TYPE: MULTIPLE_QUESTIONS