1612 CHAPTER 50. RIEMANN STIELTJES INTEGRALS

6. Let γ : [a,b]→ C be an arbitrary continuous curve having bounded variation andlet f ,g have continuous derivatives on some open set containing γ∗. Prove the usualintegration by parts formula.∫

γ

f g′dz = f (γ (b))g(γ (b))− f (γ (a))g(γ (a))−∫

γ

f ′gdz.

7. Let f (z)≡ |z|−(1/2) e−i θ2 where z= |z|eiθ . This function is called the principle branch

of z−(1/2). Find∫

γf (z)dz where γ is the semicircle in the upper half plane which

goes from (1,0) to (−1,0) in the counter clockwise direction. Next do the integral inwhich γ goes in the clockwise direction along the semicircle in the lower half plane.

8. Prove an open set, U is connected if and only if for every two points in U, there existsa C1 curve having values in U which joins them.

9. Let P,Q be two partitions of [a,b] with P ⊆Q. Each of these partitions can beused to form an approximation to V (γ, [a,b]) as described above. Recall the totalvariation was the supremum of sums of a certain form determined by a partition.How is the sum associated with P related to the sum associated with Q? Explain.

10. Consider the curve,

γ (t) ={

t + it2 sin( 1

t

)if t ∈ (0,1]

0 if t = 0.

Is γ a continuous curve having bounded variation? What if the t2 is replaced with t?Is the resulting curve continuous? Is it a bounded variation curve?

11. Suppose γ : [a,b]→ R is given by γ (t) = t. What is∫

γf (t)dγ? Explain.