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Introduction to complex analysis

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# 3.2 Further exercises

Here are some further exercises to end this section.

## Exercise 14

For each of the following functions , evaluate

where is any contour from to .

• a.

• b.

• c.

• d.

• e.

• f.

• g.

• h.

• i.

### Answer

In each case, is continuous on and has a primitive on , so we can apply the Fundamental Theorem of Calculus to evaluate the integral using any contour from to .

• a.

• b.

• c.

• d.

• e.

• f.

• g.A primitive of is

Hence

• h.A primitive of is

Hence

• i.Let , . Then and are entire (that is, and are differentiable on the whole of ), and and are entire and hence continuous. Then, using Integration by Parts (Theorem 9), we have

## Exercise 15

Evaluate the following integrals. (In each case pay special attention to the hypotheses of the theorems you use.)

• a.,

where is the arc of the circle from to passing through 1.

• b., where is as in part (a).

• c., where is the unit circle .

• d., where is the circle .

(Hint: For part (c), use the identity .)

### Answer

• a.Let , and . Then is continuous on , is a primitive of on , and is a contour in . Thus, by the Fundamental Theorem of Calculus,

• b.Let , and . Then is continuous on , is a primitive of on , and is a contour in . Thus, by the Fundamental Theorem of Calculus,

• c.The function

is continuous and has an entire primitive . Thus, by the Closed Contour Theorem,

• d.Let , and . Then is continuous on , is a primitive of on , and is a contour in . Thus, by the Closed Contour Theorem,

## Exercise 16

Construct a grid path from to in the domain of the function , for each of the following cases.

• a.,

• b.,

### Answer

The domain of is the region

• a.The figure shows one grid path in from 1 to 6 (there are many others).

• b.The figure shows one grid path in from to (again, there are many others).