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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

Write the Maclaurin series for 11+xin∑ form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms ofn as we did in Example 2 above

The series ∑n-1∞1/ns,s>1,is called the Riemann Zeta function,ζ(s). (In Problem14.2(a)you foundζ(2)=π26. Whennis an even integer, these series can be summed exactly in terms ofπ.) By computer or tables, find

(a)ζ(4)=∑n=1∞1n4(b)ζ(3)=∑n=1∞1n3(c)ζ(32)=∑n=1∞1n3/2

Derive the formula (1.4) for the sum Snof the geometric progression Sn=a+ar+ar2+.....arn-1.Hint: Multiply Snby rand subtract the result fromSn; then solve for Sn. Show that the geometric series (1.6) converges if and only if r<1; also show that if r<1, the sum is given by equation (1.8).

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

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