Complex Analysis: Does $\int_{C(0,10)} f(z)$ Equal 0?

In summary, complex analysis is a branch of mathematics that studies functions of complex numbers. The integral $\int_{C(0,10)} f(z)$ is a contour integral that allows us to calculate complex integrals using properties of complex numbers and functions. The value of $\int_{C(0,10)} f(z)$ is related to the properties of the function f(z) being integrated. It may equal 0 if f(z) is analytic within the contour, but it is still possible for it to equal 0 if f(z) has a removable singularity.
  • #1
mathmari
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Hey! :giggle:

Question 1:
If $f\in O(\Delta (0,1,15))$ then does it hold that $$\int_{C(0,10)}\frac{f(z)}{(z-6+4i)^5}\, dz=2\pi i\text{Res}\left (\frac{f(z)}{(z-6+4i)^5}, 6-4i\right )+\int_{C(0,6)}\frac{f(z)}{(z-6+4i)^5}\, dz$$ Do we maybe use here Cauchy theorem and then we get $$\int_{C(0,10)}\frac{f(z)}{(z-6+4i)^5}\, dz=0$$Question 2:
Is there a sequence of holomorphic polynomials $P_n(z), n=1,2,\ldots$ such that $$P_n(z)\rightarrow \frac{z(e^{6z}-1)(e^{4z}-1)}{\sin^2\left (\frac{z}{4}\right )\sin^2\left (\frac{2z}{5}\right )}$$ as $n\rightarrow \infty$, uniformly for $z\in \Delta (0,1,3)$ ?

since the convergence is uniform we get that $$\int \frac{z(e^{6z}-1)(e^{4z}-1)}{\sin^2\left (\frac{z}{4}\right )\sin^2\left (\frac{2z}{5}\right )}\, dz=\lim_{n\rightarrow \infty}\int P_n(z)$$ Do we have tocheck if this integral is equal to $0$ ?:unsure:
 
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  • #2
mathmari said:
Question 1:
If $f\in O(\Delta (0,1,15))$ then does it hold that $$\int_{C(0,10)}\frac{f(z)}{(z-6+4i)^5}\, dz=2\pi i\text{Res}\left (\frac{f(z)}{(z-6+4i)^5}, 6-4i\right )+\int_{C(0,6)}\frac{f(z)}{(z-6+4i)^5}\, dz$$ Do we maybe use here Cauchy theorem and then we get $$\int_{C(0,10)}\frac{f(z)}{(z-6+4i)^5}\, dz=0$$
Hey mathmari!

What is $O(\Delta (0,1,15))$? (Wondering)
 
  • #3


Hey! For question 1, yes, you can use Cauchy's theorem to show that the integral is equal to 0. For question 2, it is not necessary to check if the integral is equal to 0. The fact that the convergence is uniform means that the limit of the integral is equal to the integral of the limit. Therefore, you can just focus on showing that the polynomials converge to the given function. Hope this helps!
 

FAQ: Complex Analysis: Does $\int_{C(0,10)} f(z)$ Equal 0?

What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers and their functions. It involves the use of calculus and other mathematical tools to analyze and understand the behavior of functions defined on the complex plane.

What is the significance of the integral $\int_{C(0,10)} f(z)$?

The integral $\int_{C(0,10)} f(z)$ represents the line integral of a complex-valued function f(z) along the circular path C(0,10) on the complex plane. It is used to evaluate the total change of a function along a given path.

How is the integral $\int_{C(0,10)} f(z)$ related to the concept of contour integration?

The integral $\int_{C(0,10)} f(z)$ is a specific type of contour integral, where the path of integration is a closed curve. Contour integration is a powerful tool in complex analysis that allows us to evaluate integrals of complex functions by breaking them down into simpler parts.

Under what conditions does $\int_{C(0,10)} f(z)$ equal 0?

The integral $\int_{C(0,10)} f(z)$ equals 0 when the function f(z) is analytic (i.e. differentiable) on the entire path of integration C(0,10). This is known as Cauchy's integral theorem, which states that the integral of an analytic function along a closed path is always equal to 0.

How is complex analysis used in real-world applications?

Complex analysis has numerous applications in various fields such as physics, engineering, and economics. It is used to model and analyze physical systems, such as fluid flow and electrical circuits. In economics, it is used to study complex economic systems and make predictions. Additionally, complex analysis is also used in signal processing and image analysis.

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