Please check Cauchy Integral Formula excercise

In summary, the Cauchy Integral Formula is a fundamental theorem in complex analysis that relates the values of a complex analytic function inside a closed curve to the values of the function on the boundary of the curve. It can be derived using the Cauchy-Goursat Theorem and Cauchy's Integral Theorem, and has many important applications in mathematics and physics. It is different from the Cauchy Residue Theorem, which is used for calculating complex integrals involving singularities. Common mistakes when using the Cauchy Integral Formula include forgetting to check for singularities, using the wrong limits of integration, and not properly handling branch cuts or branch points. Careful evaluation of the function and curve is necessary to avoid these mistakes
  • #1
ognik
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2
Find $ \oint\frac{e^{iz}}{z^3}dz $ where contour is a square, center 0, sides > 1

There is an interior pole of order 3 at z=0

CIF: $ \oint\frac{f(z)}{(z-z_0)^{n+1}}dz = \frac{2\pi i}{n!} f^{(n)}(z_0) = \frac{2 \pi i}{2}f''(z_0) = -\pi i $
 
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  • #2
You can check your solution by using the Laurent expansion and looking for the coefficient of $z^{-1}$.
 

FAQ: Please check Cauchy Integral Formula excercise

What is the Cauchy Integral Formula?

The Cauchy Integral Formula is a fundamental theorem in complex analysis that relates the values of a complex analytic function inside a closed curve to the values of the function on the boundary of the curve. It is used to calculate complex integrals and has many important applications in mathematics and physics.

How is the Cauchy Integral Formula derived?

The Cauchy Integral Formula can be derived using the Cauchy-Goursat Theorem and Cauchy's Integral Theorem. These theorems state that the integral of a complex analytic function along a closed curve is equal to the sum of the function's values inside the curve. By applying these theorems to a small circle centered at a point within the curve, the Cauchy Integral Formula can be derived.

What is the difference between the Cauchy Integral Formula and the Cauchy Residue Theorem?

The Cauchy Integral Formula and the Cauchy Residue Theorem are both fundamental theorems in complex analysis, but they have different applications. The Cauchy Integral Formula is used to calculate complex integrals, while the Cauchy Residue Theorem is used to calculate complex integrals involving singularities. The Cauchy Residue Theorem is a direct consequence of the Cauchy Integral Formula.

How is the Cauchy Integral Formula used in real-world applications?

The Cauchy Integral Formula has many important applications in mathematics and physics. It is used to solve many problems involving complex integrals, such as calculating the work done by a conservative force. It is also used in the study of fluid mechanics and electromagnetism, as well as in the field of signal processing.

What are some common mistakes made when using the Cauchy Integral Formula?

Some common mistakes when using the Cauchy Integral Formula include forgetting to check for singularities within the curve, using the wrong limits of integration, and not properly handling branch cuts or branch points. It is important to carefully evaluate the given function and curve before applying the formula to avoid these mistakes.

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