Question

Let □ be the square in C with the four vertices ±2±2i. Evaluate the following integrals: (a) ∫□​z(z2+8)sin(z)​dz; (b) ∫□​(z+1)4exp(2z)​dz (c) ∫□​z2+2z​dz. (Hint. Use partial fractions.) (Do not use any version of the Cauchy Integral Formula that is more genemil than 4.4 or 5.3. When applying Cauchy's Theorem, a sketch of the relevant convex subsets is enough.)

Answer 1

To evaluate the given integrals over the complex square □, we first parameterize the square and then substitute the parameterization into the integral expressions. We simplify the expressions and evaluate the integrals using the u-substitution method.

Step-by-step explanation:

To evaluate the given integrals, we first need to parameterize the square □ in the complex plane. Let z = x + yi, where x and y are real numbers. The vertices of the square are ±2±2i, which correspond to the points (2, 2), (-2, 2), (-2, -2), and (2, -2). We can write the parameterization as:

z = 2e^(it) + 2ie^(it), where t ∈ [0, 2π]

(a) ∫□​z(z^2+8)sin(z)​dz:

Using the parameterization, we substitute z = 2e^(it) + 2ie^(it) into the integral and simplify the expression. Then we use the fact that sin(z) = (e^(iz) - e^(-iz))/2i to further simplify the integral. Finally, we evaluate the integral using the u-substitution method.

(b) ∫□​(z+1)^4exp(2z)​dz:

Using the parameterization, we substitute z = 2e^(it) + 2ie^(it) into the integral, simplify the expression, and expand (z+1)^4. Then we evaluate the integral using the u-substitution method.

(c) ∫□​z^2+2z​dz:

Using the parameterization, we substitute z = 2e^(it) + 2ie^(it) into the integral and simplify the expression. Then we evaluate the integral using the u-substitution method.