To evaluate the given integral, we will use the Cauchy-Goursat theorem which states that the integral of an analytic function over a simple closed contour is zero.

The function f(z) = cos(z) + z² + z + Im(z) is analytic everywhere in the complex plane because it is composed of elementary functions that are analytic everywhere (cosine, power, identity, and the imaginary part function).

The contour C is a triangle with vertices at z = 0, z = 1, and z = 1 + 3i. This is a simple closed contour.

Therefore, by the Cauchy-Goursat theorem, the integral of f(z) over the contour C is zero:

∫[cos(z) + z² + z + Im(z)]dz = 0

Note: The Cauchy-Goursat theorem is applicable here because the function f(z) does not have any singularities (points where the function is not defined or not analytic) inside the contour C.

Question

To evaluate the given integral, we will use the Cauchy-Goursat theorem which states that the integral of an analytic function over a simple closed contour is zero.

The function f(z) = cos(z) + z² + z + Im(z) is analytic everywhere in the complex plane because it is composed of elementary functions that are analytic everywhere (cosine, power, identity, and the imaginary part function).

The contour C is a triangle with vertices at z = 0, z = 1, and z = 1 + 3i. This is a simple closed contour.

Therefore, by the Cauchy-Goursat theorem, the integral of f(z) over the contour C is zero:

∫[cos(z) + z² + z + Im(z)]dz = 0

Note: The Cauchy-Goursat theorem is applicable here because the function f(z) does not have any singularities (points where the function is not defined or not analytic) inside the contour C.

Knowee AI · Accepted Answer

To evaluate the given integral, we will use the Cauchy-Goursat theorem which states that the integral of an analytic function over a simple closed contour is zero.

The function f(z) = cos(z) + z² + z + Im(z) is analytic everywhere in the complex plane because it is composed of elementary functions that are analytic everywhere (cosine, power, identity, and the imaginary part function).

The contour C is a triangle with vertices at z = 0, z = 1, and z = 1 + 3i. This is a simple closed contour.

Therefore, by the Cauchy-Goursat theorem, the integral of f(z) over the contour C is zero:

∫[cos(z) + z² + z + Im(z)]dz = 0

Note: The Cauchy-Goursat theorem is applicable here because the function f(z) does not have any singularities (points where the function is not defined or not analytic) inside the contour C.

Evaluate [cos(z) + z² + z + Im(z)]dz, where C is the triangle with vertices z = 0, 2 z = 1, and z = 1 + 3i.

Question

Evaluate $\cos(z) + z^2 + z + \text{Im}(z)$ dz, where C is the triangle with vertices z = 0, z = 2, z = 1, and z = 1 + 3i.

Solution

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