Use Cauchy Integral Formula to evaluate the integral

In summary, the question is asking for an evaluation of the complex integral ∫e^(e^it) dt from 0 to 2∏ using an appropriate substitution and the Cauchy Integral Formula. To solve this, we need to turn it into a contour integral, with circles being the easiest option. The parameterization z = e^{it} may be useful in this case.
  • #1
DanniHuang
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Homework Statement



The question is needed to be done by using an appropriate substitution and the Cauchy Integral Formula.

Homework Equations



Evaluate the complex integral: ∫e^(e^it) dt, from 0 to 2∏

The Attempt at a Solution


I cannot find an appropriate substitution for the integral.
 
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  • #2
Perhaps the best way to see how to do this question is that we somehow need to make this into a contour integral. The easiest things around which to integrate are circles right? For example, if f(z) is a complex function and we want to integrate around the unit circle [itex] \{ z \in \mathbb C: |z|^2 = 1 \} [/itex], it may be convenient to parameterize the circe as [itex] z = e^{it} [/itex]. Thus if somebody asks you to integrate [itex] e^{e^{it}} [/itex] for [itex] t \in [0,2\pi) [/itex], what function is being integrated about what contour?
 

1. What is the Cauchy Integral Formula?

The Cauchy Integral Formula is a mathematical theorem that allows for the evaluation of complex integrals along a closed contour. It states that if a function f(z) is analytic inside and on a simple closed contour C, then the integral of f(z) along C can be calculated using a simple formula involving the values of f(z) inside the contour.

2. How is the Cauchy Integral Formula used to evaluate integrals?

The Cauchy Integral Formula uses a contour integral to evaluate the value of a complex function at a point inside the contour. This point is known as the center of the circle of integration. The formula involves the function's values at points inside the contour as well as the contour's derivative at the center point.

3. What are the conditions for using the Cauchy Integral Formula?

The Cauchy Integral Formula can only be used for functions that are analytic inside and on a simple closed contour. This means that the function must be defined and have derivatives of all orders at every point inside the contour. Additionally, the contour must be simple, meaning it does not intersect itself or have any singular points.

4. Are there any limitations to using the Cauchy Integral Formula?

One limitation of the Cauchy Integral Formula is that it can only be used for functions that are analytic. This means that it cannot be used for functions with singularities, such as poles or branch points. Additionally, the contour must be simple and closed, which may limit its use for more complex or irregular shapes.

5. Can the Cauchy Integral Formula be used for real-valued functions?

Yes, the Cauchy Integral Formula can be used for real-valued functions as well. In this case, the contour and the function must both be real-valued, and the integral will result in a real number. However, the Cauchy Integral Formula is most commonly used for complex functions, as it is a powerful tool for solving problems in complex analysis.

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