4. Find the length of the following contours. (a) z(t)=3e^2it+2 (-pi leqslant tleqslant pi ) z(t)=e^tcost+ie^tsint (-pi leqslant tleqslant pi )

Question

Pertanyaan

4. Find the length of the following contours. (a) z(t)=3e^2it+2 (-pi leqslant tleqslant pi ) z(t)=e^tcost+ie^tsint (-pi leqslant tleqslant pi )

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Answer 1

Here's how to find the lengths of the given contours:

Understanding the Concepts

* Contour: A contour is a continuous curve in the complex plane.
* Arc Length Formula: The length of a contour defined by a complex function *z(t)* from *t = a* to *t = b* is given by:

```
L = ∫[a, b] |z'(t)| dt
```

where *z'(t)* is the derivative of *z(t)* with respect to *t*.

Solving the Problems

(a) z(t) = 3e^(2it) + 2

1. Find the derivative:
* z'(t) = 6ie^(2it)

2. Calculate the magnitude of the derivative:
* |z'(t)| = |6ie^(2it)| = 6|e^(2it)| = 6 (since |e^(2it)| = 1)

3. Integrate to find the arc length:
* L = ∫[-π, π] 6 dt = 6t |[-π, π] = 12π

Therefore, the length of the contour z(t) = 3e^(2it) + 2 is 12π.

(b) z(t) = e^t cos(t) + ie^t sin(t)

1. Find the derivative:
* z'(t) = e^t(cos(t) - sin(t)) + ie^t(sin(t) + cos(t))

2. Calculate the magnitude of the derivative:
* |z'(t)| = √[(e^t(cos(t) - sin(t)))² + (e^t(sin(t) + cos(t)))²]
* |z'(t)| = √[e^(2t)(cos²(t) - 2cos(t)sin(t) + sin²(t) + sin²(t) + 2sin(t)cos(t) + cos²(t))]
* |z'(t)| = √[e^(2t)(2)] = √2 * e^t

3. Integrate to find the arc length:
* L = ∫[-π, π] √2 * e^t dt = √2 * e^t |[-π, π] = √2(e^π - e^(-π))

Therefore, the length of the contour z(t) = e^t cos(t) + ie^t sin(t) is √2(e^π - e^(-π)).

Pertanyaan

4. Find the length of the following contours. (a) z(t)=3e^2it+2 (-pi leqslant tleqslant pi ) z(t)=e^tcost+ie^tsint (-pi leqslant tleqslant pi )

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