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Chapter 14: Q27P (page 673)

Using the definition (2.1) of , show that the following familiar formulas hold. Hint : Use the same methods as for functions of a real variable.

27..

Short Answer

Expert verified

It is proved that the derivative is,

.

Step by step solution

01

To write the definition (2.1) of derivative of function

The definition of the derivative of the functionis as follows:

The derivative of the function is defined as,

Where, .

02

To write the product of functions

Let be the function.

By using the definition of differentiation,

Thus,

Using product of limits, above equation becomes,

03

To find the derivative

Adding and subtracting the term to the numerator of ,

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Most popular questions from this chapter

(a) Show that if f(z)tends to a finite limit as z tends to infinity, then the residue of f(z) at infinity is.

(b) Also show that iff(z)tends to zero as z tends to infinity, then the residue of f(z) at infinity is -limz2f'(z).

Evaluate the following integrals by computing residues at infinity. Check your answers by computing residues at all the finite poles. (It is understood that 鈭� means in the positive direction.)

鈭�1-z21+z2dzzaround |z|=2

To find that the integrals by computing residue at infinity.

鈭�cz2(2z+1)(z2+9) around |z|=5 .

Find the real and imaginary parts u(x,y) andv(x,y) of the following functions.

cosz炉

To prove that the sum of the residues at finite points plus the residence at infinity is zero.
See all solutions

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