Evaluate the contour integrals along the circular paths indicated ∫(2z + 3)dz

General
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Evaluate the contour integrals along the circular paths indicated ∫(2z + 3)dz

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Evaluate the contour integrals along the circular paths indicated ∫(2z + 3)dz

Answer: To evaluate a contour integral like \int (2z + 3) \, dz along a circular path, you can apply complex analysis techniques, specifically focusing on the nature of the integrand.

Steps to Solve:

  1. Understand the Context:

    • The integral \int (2z + 3) \, dz represents a line integral of a complex function along a certain path in the complex plane.

  2. Consider the Given Path:

    • If the path is circular, say |z - a| = R, where a is the center and R is the radius, it’s important to examine if the function is analytic (holomorphic) on and inside this path.

  3. Check Analyticity:

    • The function 2z + 3 is a polynomial in z, which means it is analytic everywhere in the complex plane.

  4. Apply Cauchy’s Theorem:

    • Since 2z + 3 is analytic everywhere, the integral over a closed curve (like a circle) will be zero due to Cauchy’s theorem. This is a direct result because there are no singularities inside the path.

  5. Evaluate the Integral:

    • Based on Cauchy’s theorem:
    \int_{|z-a|=R} (2z + 3) \, dz = 0

    because the function is analytic over the path and its enclosed region.

Summary:

When evaluating the contour integral of an analytic function like 2z + 3 over a closed, simple path such as a circle, the result is zero due to Cauchy’s theorem. Analytic functions without singularities inside the contour lead to integrals equaling zero for closed paths.

If further details about the path were needed or if there were singularities, additional methods might be required.

Feel free to ask more questions if you need further clarification! @anonymous2