Integral of f(z) dz Around C1 & C2: Complex Math Solutions

In summary, the conversation is about finding the integrals of a function involving sin(z) around two different circles. The suggestion is to use the Cauchy Integral Formula or the Residue Theorem, and the values obtained are 2pi i and 4pi^2 i. There is also a discussion about the function being analytic and having a residue at 0.
  • #1
squenshl
479
4

Homework Statement


We know sin(z) has zeros at integral multiples of pi. Let f(z) = z2/sin2(z)
How do I find the integral of f(z) dz around C1 (C1 is the circle |z| = 1 orientated anti-clockwise) and how do I find the integral of f(z) dz around C2 (C2 is the circle |z - pi| = 1 orientated anti-clockwise).

Homework Equations





The Attempt at a Solution


Do I use the Cauchy Integral formula for these integrals.
If not, how would I go about doing these.
 
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  • #2
how about thinking residues?
 
  • #3
I got 1 as my integral.
 
  • #4
No. 1 is my value of f(z) (using L'Hopitals rule and the fact that f(z) has a removable singularity at z = 0 so this function is analytic).
My limits of integration are 0 and 2pi so my integral is 2pi.
 
  • #5
No again. We use the residue theorem
integral = 2 pi i (sum of the residues)
= 2 pi i (1)
= 2 pi i
and for the next integral I got 4 pi^2 i
 
Last edited:
  • #6
If your function is analytic how can it have a residue at 0?
 

FAQ: Integral of f(z) dz Around C1 & C2: Complex Math Solutions

1. What is the integral of f(z) dz around C1 and C2?

The integral of f(z) dz around C1 and C2 is a complex math solution that calculates the sum of the infinitesimal contributions of f(z) along the path defined by C1 and C2. In other words, it is the area under the curve of f(z) around the closed curves C1 and C2.

2. How is the integral of f(z) dz around C1 and C2 calculated?

The integral of f(z) dz around C1 and C2 is calculated by using the Cauchy Integral Formula, which states that the integral of f(z) dz around a closed curve C is equal to 2πi multiplied by the sum of the residues of f(z) at all the singularities enclosed by C. This formula can be applied to calculate the integral of f(z) dz around C1 and C2.

3. What is the significance of calculating the integral of f(z) dz around C1 and C2?

The integral of f(z) dz around C1 and C2 has many applications in mathematics, physics, and engineering. It is used to solve complex differential equations, evaluate complex line integrals, and calculate important physical quantities such as work, displacement, and electric charge distribution. It also helps in understanding the behavior of functions in the complex plane.

4. Can the integral of f(z) dz around C1 and C2 be calculated for any function?

Yes, the integral of f(z) dz around C1 and C2 can be calculated for any complex function that is analytic within the closed curves C1 and C2. This means that the function must have a continuous first derivative at all points within the curves. If the function is not analytic, then the integral cannot be calculated using the Cauchy Integral Formula.

5. How is the integral of f(z) dz around C1 and C2 related to the concept of contour integration?

Contour integration is a method used to evaluate complex integrals by transforming them into line integrals along a closed curve. The integral of f(z) dz around C1 and C2 is an example of contour integration, where the path C is defined by the curves C1 and C2. This method is particularly useful for solving complex integrals that cannot be evaluated using traditional methods.

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