Complex trigonometric integral

In summary, the task is to calculate the complex integral along the closed path |z-2i|=2 for the function sin(z)/(z^2+π^2), using calculus of residues. The denominator has singularities at z=±iπ, which lie within the contour. Therefore, the integral can be calculated using the formula (2πi) x Res[f(z), iπ].
  • #1
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Homework Statement


Calculate the complex integral along the closed path indicated:
$$ \oint_C\frac{\sin{z}}{z^2+\pi^2}dz,\,\,|z-2i|=2.$$

Homework Equations


$$ \sin{z}=\frac{e^{iz}-e^{-iz}}{2i} $$
$$ e^{iz}=e^{i(x+iy)}=e^{-y+ix}=e^{-y}(\cos{x}+i\sin{x}) $$

The Attempt at a Solution


I really don't know what to do here.. Everything I tried led me to a dead end. Is there a clever substitution to be made? I tried substituting ##z=x+iy##, I tried ##z=e^{it}+2i## and even tried expanding ##\sin{z}##, but it got me nowhere. Any help is appreciated. Thanks!
 
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  • #2
You should use calculus of residues!
 
  • #3
The denominator ##z^2 + \pi^2## has singularities at ##z = \pm i \pi##.

Do these lie within the positively oriented contour ##|z - 2i| = 2##?
 
  • #4
Zondrina said:
The denominator ##z^2 + \pi^2## has singularities at ##z = \pm i \pi##.

Do these lie within the positively oriented contour ##|z - 2i| = 2##?

Yes, ##i\pi## lies within the contour, which means that the integral for any closed path around ##i\pi## would wield the right answer, but that didn't help me much. I still don't know how to calculate the integral.
 
  • #6
$$\oint_C f(z) \space dz = (2 \pi i) \times \space \text{Res}[f(z), i \pi]$$
 

Related to Complex trigonometric integral

1. What is a complex trigonometric integral?

A complex trigonometric integral is an integration that involves complex-valued trigonometric functions, such as sine, cosine, and tangent. These integrals can be more challenging to solve than real-valued integrals because they involve complex numbers and require special techniques.

2. How do you solve a complex trigonometric integral?

To solve a complex trigonometric integral, you need to use techniques such as substitution, integration by parts, or trigonometric identities. These techniques help simplify the integral and make it easier to solve. Additionally, you may need to use complex analysis methods, such as contour integration.

3. What are some common examples of complex trigonometric integrals?

Some common examples of complex trigonometric integrals include integrals involving exponential functions, inverse trigonometric functions, or multiple trigonometric functions. For example, ∫sin(z)cos(z)dz or ∫tan(z)sec(z)dz.

4. What are the applications of complex trigonometric integrals?

Complex trigonometric integrals have many applications in physics, engineering, and mathematics. They are used to solve problems involving oscillatory systems, electromagnetism, and signal processing. Additionally, they are essential in solving differential equations and evaluating complex-valued functions.

5. Are there any tips for solving complex trigonometric integrals?

Some tips for solving complex trigonometric integrals include using trigonometric identities to simplify the integral, carefully choosing a substitution, and using symmetry properties of the integral. It is also helpful to have a good understanding of complex numbers and their properties. Practice and patience are key to becoming proficient in solving complex trigonometric integrals.

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