Cauchys Integral Formula Calculator

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Example

Created on 2024-06-20Asked by Jacob Wright (Solvelet student)

Use Cauchy's Integral Formula to find

f'(0)

for

f(z) = \frac{e^z}{z^2+1}

with a contour

|z| = 1

Solution

Step 1: Identify the function

f(z) = \frac{e^z}{z^2+1}

and the contour

|z| = 1

. Step 2: Use Cauchy's Integral Formula for the derivative:

f'(a) = \frac{1}{2\pi i} \int_{\gamma} \frac{f(z)}{(z-a)^2} \, dz.

Step 3: Since

a = 0

, the formula becomes:

f'(0) = \frac{1}{2\pi i} \int_{|z|=1} \frac{e^z}{z^2+1} \frac{1}{z^2} \, dz.

Step 4: Simplify the integral:

f'(0) = \frac{1}{2\pi i} \int_{|z|=1} \frac{e^z}{z^2 (z^2 + 1)} \, dz.

Step 5: By residue theorem, evaluate the residue at

z = 0

\text{Res}\left( \frac{e^z}{z^2 (z^2 + 1)}, 0 \right) = \frac{d}{dz} \left. \left( \frac{e^z}{z^2 + 1} \right) \right|_{z=0} = \frac{e^0}{0^2 + 1} = 1.

Step 6: Conclusion. The value of

f'(0)

is 1. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Theodore Walker on Solvelet

1. Use Cauchy's integral formula to evaluate the integral

\oint_C \frac{z^3 + 2z}{z - i} \, dz

, where

C

is the unit circle,2. Calculate the derivative

g(x) = e^{2x^3 - 4x}

DefinitionI have used Cauchys integral formula, one of the most important theorems in complex analysis which gives rise to a deep correspondence between the values of a complex function inside a region and its values on the boundary of the region. It says that the value of a function at any point within a simply connected region in which it is analytic and on its boundary can be represented by (an) integral(s) involving the values of the function on the boundary. Applications of Cauchy's formula are found in complex analysis, physics, engineering, and signal processing. Example: Cauchys integral formula in electrical engineering: This is specifically used in the analysis of circuits with distributed parameters through a process of transformation from the time domain to the frequency domain using the Laplace transform.

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