Although Corollary 2 does not apply to the function 1 / (z — z_0) in the plane punctured at z_0, Theorem 6 can be used as follows to show that
(see attached 1)
for any circle C traversed once in the positive direction surrounding the point z_0. Introduce a horizontal branch cut from z_0 to infinity as in Fig. 4.25. In the resulting "slit plane" the function 1 / (z — z_0) has the anti-derivative Log (z — z_0). Apply Theorem 6 to compute the integral along the portion of C from alpha to beta as indicated in Fig. 4.25. Now let alpha and beta approach the point t on the cut to evaluate the given integral over all of C.
(see attached 1)
Theorem 6. Suppose that the function f (z) is continuous in a domain D and has an anti-derivative F(z) throughout D; i.e., dF(z)Idz = f (z) for each z in D. Then for any contour gamma lying in D, with initial point z_I and terminal point z_T, we have
(see attached 2).

a) Let M>0, and let f:[a,b]-->R be a function which is continuous on [a,b] and differentiable on (a,b), and such that |f'(x)| <= M for all x belonging to (a,b) (derivative of f is bounded). Show that for any x,y belonging to [a,b] we have the inequality |f(x)-f(y)| <= M|x-y|. *apply mean value theorem.
b) Let f:R-->R be a di

Show that the derivative of g(z) exists, and hence g(z) is complex analytic, at points where g(z) does not diverge, thereby proving that the points where g(z) diverges are the only singularities.
g(z)= eiz/ z2 +4z+5

Determine whether Rolle's Theorem can be applied to f on the closed interval [a,b]. If Rolle's theorem can be applied, find all values of c in the open interval (a,b) such that f'(c) = 0.
f(x) = sin x, [0, 2pi]

Assume that f is differentiable for each x and there exists M>0
such that
for each x
Prove that f is uniformly continuous on D.
Hint: Can use the mean value theorem.
keywords: differentiability, continuity

(1) let f:C----R be an analytic function such that f(1)=1. Find the value of f(3)
(2) Evaluate the integral over & of dz/ z^2 -1 where & is the circle |z-i|=2
(3)Evaluate the integral over & of (z-1/z) dz where & is the line path from 1 to i
(4) Evaluate the integral between 2pi and 0 of
e^-i@ . e ^e^i@ d@

Use part I of the Fundamental Theorem of Calculus to find the derivatives of the following functions; answers must use correct variable.
a. f(x)=the integral as pi goes to x of (1+cos[t])dt; f'(x)=___
b. f(u)=the integral as -1 goes to u of [1/(x+4x^2)]dx; f'(u)=___

Apply the given theorem to verify that each of these functions is entire:
(a) f (z) = 3x + y + i (3y - x)
(b) f (z) = sin x cosh y + i cos x sinh y
(c) f (z) = e-y sin x - i e-y cos x
(d) f (z) = (z2 - 2) e-x e-iy.
See attached file for proper formatting.

Please explain how to prove the following. As much explanation as possible would be great
Let p(x) = Ax^2 + Bx + C. Prove that for any interval [a,b], the value c guaranteed by the Mean Value Theorem is the midpoint of the interval.

Let f(x) be a continuous function of one variable.
a) Give the definition of the derivative.
b) Use this definition to find the derivative of f(x)=x^2+2x-5
c) Evaluate f'(2)