text: 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.
explanation: The document contains a math problem related to the Cauchy-Goursat Theorem. It provides a problem statement with two parts, each requiring the integration of a function and the application of the theorem. The document also includes a solution summary, indicating that the Cauchy-Goursat Theorem is utilized.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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distilabel_metadata.content: 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem related to the Cauchy-Goursat Theorem. It provides a problem statement with two parts, each requiring the integration of a function and the application of the theorem. The document also includes a solution summary, indicating that the Cauchy-Goursat Theorem is utilized.
  
  Label: question-and-answer