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Chapter 6: Problem 14

Find \(\int_{C_{3}^{+}(1)}\left(z^{2}+1\right)^{-2} d z\).

Short Answer

Expert verified
The integral is 0 by the Cauchy Integral Theorem.

Step by step solution

01

Understand the problem

We need to find the integral of the function \((z^2 + 1)^{-2}\) over the positively oriented circle centered at 0 with radius 1 in the complex plane, denoted as \(C_3^{+}(1)\). This suggests using methods from complex analysis.
02

Identify the function's singularities

The function \((z^2 + 1)^{-2}\) has singularities where \(z^2 + 1 = 0\). This simplifies to \(z = i\) and \(z = -i\). Both points are singularities, but we must focus on whether they lie inside the contour.
03

Determine singularities inside the contour

The contour is a circle of radius 1 centered at the origin. Since \(|i| = 1\) and \(|-i| = 1\), both \(i\) and \(-i\) are on the contour boundary, not inside. Thus, there are no singularities strictly inside the contour.
04

Apply Cauchy's Integral Theorem

Since there are no singularities inside the contour \(C_3^{+}(1)\), the Cauchy Integral Theorem states that the integral of a holomorphic function over such a closed contour is 0. The function \((z^2 + 1)^{-2}\) is meromorphic with no poles inside the contour. Therefore, \[\int_{C_{3}^{+}(1)} (z^2 + 1)^{-2} \, dz = 0.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Contour Integration
Contour integration is a technique from complex analysis used to evaluate integrals of complex functions along a specific path or contour in the complex plane. Imagine drawing a path in this world of real and imaginary numbers. This path indicates where and how we should integrate the function.
For the problem \((z^2 + 1)^{-2}\), the contour is a circle of radius 1 centered at the origin. This is noted as \(C_3^{+}(1)\), a positively oriented contour, meaning we travel along the path counter-clockwise.
  • Choosing the Contour: The contour must be closed, like a loop, so you end where you started.
  • Contour Properties: The path and the function should be smooth and properly defined along the entire path.
By carefully selecting and understanding our contour, it becomes easier to apply complex analysis theorems, such as Cauchy's Integral Theorem.
Cauchy's Integral Theorem
Cauchy's Integral Theorem is a central statement in complex analysis. It tells us that if a function is holomorphic (complex-differentiable) across a contour and its interior, the integral over that contour is zero.
In our problem, we identify that \( (z^2 + 1)^{-2} \) is meromorphic with no poles inside the contour \(C_3^{+}(1)\). Since there are no singularities within this circle, the function is holomorphic there, allowing us to use this theorem.
  • Key Condition: The function must be holomorphic in the region bounded by and on the contour.
  • Result: If the function satisfies the condition, then \(\int_C f(z) \, dz = 0\). This simplifies many complex integral problems to zero.
This theorem forms a backbone for more complex theorems and results, building upon it to solve difficult complex analysis integrals.
Singularities
Singularities in complex analysis are special points where a function becomes undefined or behaves "strangely."
For the function \(f(z) = (z^2 + 1)^{-2}\), singularities occur where the denominator equals zero: \(z^2 + 1 = 0\). Solving gives us \(z = i\) and \(z = -i\). However, due to the contour's nature, we need to confirm if these points fall inside it.
  • Type of Singularities: A singularity could be a pole, essential singularity, or removable singularity.
  • Within the Contour: Only singularities inside the contour affect the integral value. Here, both \(i\) and \(-i\) are on the boundary, not inside.
Knowing about singularities helps decide if Cauchy's Integral Theorem can be applied. If there are singularities within, different techniques like residue calculus might be needed.

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Most popular questions from this chapter

Evaluate \(\int_{C}^{z} \exp z d z\), where \(C\) is the square with vertices \(0,1,1+i\), and \(i\) taken with the counterclock wise orientation.

Evaluate \(\int_{C} \exp z \mathrm{~d} s\), where \(C\) is the straight-line segment joining 1 to \(1+i \pi\).

Find \(\int_{C} z^{-2}\left(z^{2}-16\right)^{-1} \exp z d z\) along the following contours: (a) The circle \(C_{1}^{+}(0)\). (b) The circle \(C_{1}^{+}(4)\).

Find \(\int_{C}\left(z^{2}-z\right)^{-1} d z\) for (a) circle \(C=C_{2}^{+}(1)=\\{z:|z-1|=2\\}\) having positive orientation. (b) circle \(C=C_{j}^{+}(1)=\left\\{z:|z-1|=\frac{1}{2}\right\\}\) having positive orientation.

Sketch the following curves. (a) \(z(t)=t^{2}-1+i(t+4)\), for \(1 \leq t \leq 3\) (b) \(z(t)=\sin t+i \cos 2 t\), for \(-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}\). (c) \(z(t)=5 \cos t-i 3 \sin t\), for \(\frac{\pi}{2} \leq t \leq 2 \pi\).
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