Question

Use Cauchy's theorem or integral formula to evaluate the integrals. $$\oint_{C} \frac{\sin 2 z d z}{6 z-\pi} \text { where } C \text { is the circle }|z|=3$$

Short answer

Using Cauchy’s integral formula, the integral evaluates to \[ \frac{\frac{√3}{2}}{\frac{\text{π}}{6}} = \frac{\text{π} \text{√3i}}{4} \]

Step-by-step solution

Recognize the Cauchy's integral formula

Cauchy's integral formula states that if a function is analytic and the path of integration is a simple closed curve, you can compute the integral of the function over that curve.

Identify the integral form

The given integral is \[ \frac{1}{2\text{π}i} \frac{f(z)}{z-a} \text{ where } f(z) = \text{sin}(2z) \text{ and } a = \frac{\text{π}}{6} . \]

Confirm the point lies inside the contour

Verify that the singularity point \( a = \frac{\text{π}}{6} \) lies within the contour \( |z| = 3 \), which is indeed true because \( | \frac{\text{π}}{6} | < 3 \).

Apply Cauchy’s integral formula

Following the integral formula, we have\[ f(a) = \frac{f(z)}{2\text{π}i} \frac{dz}{z-\frac{\text{π}}{6}} = \text{sin}\bigg({2 \times \frac{\text{π}}{6}}\bigg) = \text{sin}(\frac{\text{π}}{3} = \frac{\text{√3}}{2} \]

Substitute the values

Using formula we now substitute our function and integral constant values,\[ \text{sin} \bigg(2× (\frac{\text{π}}{6}\bigg) = \text{sin}(\frac{\text{π}}{3}) = \frac{\text{√3}}{2} \]

Calculate the integral

Now calculate following the respective integral constant\[ \frac{2\text{π}i}{6z-(π/6}= \frac{\text{πi}}{3} + (\text{√3}/{2}) \]

Key concepts

Complex Analysis

Complex Analysis is a fascinating field of mathematics that deals with functions of complex variables. It extends the concept of calculus to the complex plane involving real and imaginary numbers. The main ideas of complex analysis include:

• **Complex functions**: Functions that take and return complex numbers. These include polynomials, exponentials, and trigonometric functions where the input is a complex number.


• **Analyticity**: A function is called analytic if it is differentiable at every point in its domain. This means the function not only has a derivative but the derivative itself is a function that's also differentiable.

Complex analysis has various applications such as evaluating complex integrals, solving differential equations, and even in engineering fields like fluid dynamics and electrical engineering. Understanding the basics of complex analysis is crucial for grasping topics like Cauchy's Integral Theorem and Cauchy's Integral Formula.

Contour Integration

Contour Integration is a technique used in complex analysis to evaluate certain types of integrals involving complex functions. It's particularly useful when dealing with integrals over paths in the complex plane.

Key points include:

• **Closed Path**: When we talk about contour integration, we often refer to integrals over a closed path (or contour) in the complex plane. A closed path means starting and ending at the same point.


• **Cauchy's Integral Theorem**: This theorem states that if a function is analytic inside and on some closed contour, then the integral of the function around this contour is zero.


• **Cauchy's Integral Formula**: This is an extension of Cauchy's theorem and provides a way to directly evaluate certain types of integrals. Specifically, for an analytic function inside and on a simple closed contour, \[ f(a) = \frac{1}{2\text{π}i} \frac{\text{∮f(z)dz}}{z-a} \] where \(a\) is a point inside the contour.

In our problem, we used Cauchy's Integral Formula to evaluate the integral of a given function by ensuring the singularity lies within the contour and applying the formula to find the result.

Analytic Functions

Analytic Functions are central to complex analysis. They exhibit smoothness and are infinitely differentiable within their domain.

• **Definition**: A function \(f(z)\) of a complex variable \(z\) is said to be analytic at a point if it has a complex derivative at every point in some neighborhood of that point.


• **Holomorphic Functions**: Another name for analytic functions. These functions not only have derivatives but those derivatives are also continuous.


• **Power Series Representation**: If a function is analytic in a region, it can be represented by a power series in that region. This property is incredibly powerful and not shared by real-variable functions.


• **Applications**: They appear in solving physical problems, engineering, and even in complex polynomial equations.

In the context of the given problem, the function \( \text{sin} 2z \) is analytic over the contour \( |z| = 3 \). By confirming analyticity and using Cauchy's Integral Formula, we can effectively evaluate the integral.