Evaluate the Integrals (3 Problems)

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Integrals of Complex-Valued Functions

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Transforming Representations

Derive the equation of the line through the points ... and ... in the ... plane, shown in Fig. 37. Then use it to find the linear function ... which can be used in equation (9), Sec. 38. to transform representation (2) in that section into representation (10) there.... The parametric representation used For any given arc C is ...continues

Analytic Function : Chain Rule And Cauchy-Riemann Equation

Suppose that a function f(z) is analytic at a point z0 = z(t0) lying on a smooth arc.... Please see the attached file for the fully formatted problems.

Integral of a Semicircle and Segment

f(z) = z - 1 and C is the arc from z = 0 to z = 2 consisting of (a) the semicircle z = 1 - e^(iθ) (pi ≤ θ ≤ 2pi) (b) the segment 0 ≤ x ≤ 2 of the real axis. Find the integral ∫c f(z) dz for the two cases.

Solve the Complex Integrals

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Integral Solved Using Principal Value and Principal Branch of an Integrand

Use parametric representation in exercise 10 for the oriented circle C0 there to show that....where a is any real number other than zero and where the principal branch of the integrand and where the principal value of R^G are taken. Please see the attached file for the fully formatted problems.

Analysis of a Midpoint of a Line Segment

Let C denote the line segment from z = i to z= 1. By observing that, of all the points on that line segment, the midpoint is the closest to the origin, show that |∫c dz/z^4| ≤ 4 sqrt(2) without evaluating the integral. Please see the attached file for the fully formatted problems.

Circle Desciribed in a Counterclockwise Direction : Find Limit of Integral Using L'Hopital's Rule

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Boundary of a Square : Analysis with Trigonometric and Hyperbolic Functions

Let CN denote the boundary of the square formed by the lines... Please see the attached file for the fully formatted problems.