- #1
Cauchy.riemann integral theorem or formula
In summary: That looks good. For (ii) you should mention that you have changed the path without crossing z=0.How have I changed the integral path? You mean because I integrate it from 0 to 2pi?In summary, the conversation discusses different methods for finding integrals and the importance of knowing if the point 0 is inside or outside the given circles. There is also mention of the pole at z=0 for 1/z and how the integration path can be changed without crossing z=0. It is noted that the integral will have different values depending on whether or not z=0 is inside the circle. The conversation also mentions Cauchy's integral theorem and the use of parametrization for calculating integrals
- #2
lurflurf
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There are many ways to find the integrals. The key point is if 0 is inside the given circles of outside.
What is the value in each case (0 is inside and 0 is outside)?
For C1 and C2 is 0 inside or outside?
What is the value in each case (0 is inside and 0 is outside)?
For C1 and C2 is 0 inside or outside?
- #3
MissP.25_5
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lurflurf said:
The first one : 0 is outside the circle while the second one is inside. Am I right? Why do we have to know this?
- #4
lurflurf
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^You do not have to know, but it helps. Do the integral for any circle centered at the origin. What is the value? That will be the same value as any closed contour that has 0 inside it.
1/z has a pole at z=0
Integrating around it will not be zero.
I am not sure how far into your class you are. That might be a new fact.
1/z has a pole at z=0
Integrating around it will not be zero.
I am not sure how far into your class you are. That might be a new fact.
- #5
MissP.25_5
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lurflurf said:
Yes, I have learned that 1/z has a pole at z=0 but I am not sure what it means. Do you have a diagram so I can see it?
- #6
MissP.25_5
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lurflurf said:
So, the for no.1, 1/z has a pole at z=0 because the circle is outside of 0? What about the second one? 0 is inside the circle, does that mean the pole is not at z=0?
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- #7
- #8
lurflurf
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^That looks good. For (ii) you should mention that you have changed the path without crossing z=0.
The pole (for 1/z) is always at z=0. The important thing is to know if z=0 is inside the circle or not. As you have said it is not for (i) and it is for (ii). Again I do not know what theorems and methods you have covered so far. You can calculate the two integrals by parametrization for example. The integration path can be moved without changing the value of the integral so long as the path does not cross any points where the function is not differentiable such as z=0 for 1/z. Thus the integral around a circle will have one value if z=0 is inside the circle and another value if z=0 is outside the circle.
The pole (for 1/z) is always at z=0. The important thing is to know if z=0 is inside the circle or not. As you have said it is not for (i) and it is for (ii). Again I do not know what theorems and methods you have covered so far. You can calculate the two integrals by parametrization for example. The integration path can be moved without changing the value of the integral so long as the path does not cross any points where the function is not differentiable such as z=0 for 1/z. Thus the integral around a circle will have one value if z=0 is inside the circle and another value if z=0 is outside the circle.
- #9
MissP.25_5
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lurflurf said:
How have I changed the integral path? You mean because I integrate it from 0 to 2pi?
FAQ: Cauchy.riemann integral theorem or formula
What is the Cauchy-Riemann integral theorem?
The Cauchy-Riemann integral theorem is a fundamental theorem in complex analysis that relates the integral of a complex-valued function over a closed curve to the values of the function inside the curve. It is named after the mathematicians Augustin-Louis Cauchy and Bernhard Riemann.
How is the Cauchy-Riemann integral theorem used?
The Cauchy-Riemann integral theorem is used to calculate the value of a complex integral, which can be used to solve a variety of problems in mathematics and physics. It is also a key tool in the study of analytic functions, which are functions that can be expressed as a power series and have a number of useful properties.
What is the Cauchy-Riemann formula?
The Cauchy-Riemann formula is a specific application of the Cauchy-Riemann integral theorem, which states that for a complex-valued function f(z) = u(x,y) + iv(x,y), where u and v are real-valued functions of the variables x and y, the following conditions must hold for the function to be analytic:
- The partial derivatives of u and v with respect to x and y must exist and be continuous
- The partial derivatives of u with respect to x must equal the partial derivatives of v with respect to y
- The partial derivatives of u with respect to y must equal the negative of the partial derivatives of v with respect to x
What is the relationship between the Cauchy-Riemann integral theorem and Cauchy's integral theorem?
The Cauchy-Riemann integral theorem is a generalization of Cauchy's integral theorem, which states that the integral of a function over a closed curve in the complex plane is equal to the sum of its residues inside the curve. The Cauchy-Riemann integral theorem expands on this by allowing for more complex functions and integrating over a larger class of curves.
What are some real-world applications of the Cauchy-Riemann integral theorem?
The Cauchy-Riemann integral theorem has a variety of applications in fields such as physics, engineering, and economics. It is used to calculate the value of complex integrals in the study of fluid dynamics, electromagnetic theory, and quantum mechanics. It also has applications in signal processing and financial mathematics.