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Chapter 1: Q29P (page 1)

Using the definition (2.1) of show that the following familiar formulas hold.

Short Answer

Expert verified

By using the definition 2.1, it is showed that the following familiar formula hold.

Step by step solution

01

Definition 2.1 of analytic function.

A function is analytic (or regular or holomorphic or monogenic) in a region of the complex plane if it has a (unique) derivative at every point of the region. The statement is analytic at a point means that has a derivative at every point inside some small circle about .

The derivative of is defined (just as it is for a function of a real variable) by the equation,

Where; and .

02

Apply definition 2.1.

Given the function is:

By the definition 2.1,

Solve the function is as follows:

Here it is clear that the following function hold by using the definition 2.1.

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Most popular questions from this chapter

Show that if p is a positive integer, thenpn=0when n>p , so (1+x)p=∑pnxnis just a sum ofp+1terms, from n=0 to n=p . For example,(1+x)2has 3 terms,(1+x)3 has4terms, etc. This is just the familiar binomial theorem.

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

limx→0sin22xx2 .

Test the following series for convergence.

2.∑n=1∞(−2)nn2

In the following problems, find the limit of the given sequence asn→∞

10nn!

Show thatis the distance between the points and in the complex plane. Use this result to identify the graphs in Problems 53,54,59,and60without computation.
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