180k views
2 votes
Evaluate ∫eᶻ-1/e dz along the unit circle___

1 Answer

7 votes

The value of the integral ∫eᶻ-1/e dz along the unit circle is 2πi.

We can evaluate the integral ∫eᶻ-1/e dz along the unit circle using the Cauchy integral formula.

Here's the approach:

Identify the singularities:

The integrand eᶻ-1/e has a singularity at z = 0. Since the unit circle encloses z = 0, we need to evaluate the residue of the integrand at this point.

Residue theorem:

The Cauchy integral formula states that for a function f(z), analytic within and on a simple closed contour C except for a finite number of poles a1, a2, ..., an inside C, where f(z) is also finite, we have:

∫C f(z) dz = 2πi ∑ni=1 Res(f, ai)

where Res(f, ai) is the residue of f(z) at the pole a_i.

Evaluate the residue:

The residue of eᶻ-1/e at z = 0 is 1. This can be found using various methods, such as Laurent series expansion or direct differentiation.

Apply the Cauchy integral formula:

Using the residue theorem, we get:

∫eᶻ-1/e dz = 2πi * Res(eᶻ-1/e, 0) = 2πi * 1 = 2πi

Therefore, the value of the integral ∫eᶻ-1/e dz along the unit circle is 2πi.

User Shamster
by
8.3k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.