Integrating Sin(1/z) and Z Sin (1/Z^3) Over C (Circle of Radius 1)

In summary, The conversation is discussing the integration of two functions over a circle with a radius of 1 centered at 0. The person is having trouble with the answer always being infinity and is seeking help with finding the poles and their residues. The suggestion is to use the Laurent series for sin(z) and replace z with 1/z and 1/z^3 to make the integrals easier.
  • #1
fahd
40
0
hi there
im confused with this question..
Integrate 1) Sin(1/z) dz
and 2) Z sin (1/Z^3)

where Z is any complex number., over C which is a circle of radius 1 centred at 0

i tried using the cauchy integral formula and stuff but somehow the answer always comes infinity...is that right ..please help..got exam tomorrow!
 
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  • #2
Where are the poles and what are its residues at each pole?
 
  • #3
Use Laurent series
 
  • #4
Presumably, you know the Taylor's series for sin(z). Replace z in that series by 1/z and 1/z3 to get the Laurent Series for sin(1/z) and sin(1/z3) respectively. Then the integrals should be easy.
 

1. What is the definition of the given function?

The function given is f(z) = sin(1/z) + z*sin(1/z^3) integrated over the complex plane with a radius of 1.

2. What is the domain and range of the function?

The domain is the entire complex plane except for z = 0, and the range is the set of all complex numbers.

3. How is this function integrated over the complex plane of radius 1?

This function can be integrated using complex calculus techniques, specifically the Cauchy integral formula or the residue theorem.

4. What is the significance of the circle of radius 1 in this integration?

The circle of radius 1 is significant because it defines the boundary of the integration and helps to determine the behavior of the function at infinity.

5. What are the possible applications of integrating this function over the complex plane?

This type of integration has applications in various areas of mathematics and physics, such as complex analysis, differential equations, and quantum mechanics.

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