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Chapter 14: Q5P (page 667)

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

Rez

Short Answer

Expert verified

The real part ux,yof the function is x , and the imaginary part vx,yof the function is 0.

Step by step solution

01

Definition of complex numbers

Complex numbers are expressed in the form of z=x+iy , where x,y are real numbers, and i is an imaginary number.

Similarly, the function of z is represented as follows:

f(z)=f(x+iy)=u(x,y)+iv(x,y)

, whereu(x,y) is the real part andv(x,y) is the imaginary part.

02

Substitute the value in the given function.

The given function isRez .

Substitutez=x+iy in Rezas follows:

Rez=Rex+iy=x=x+0i

The above equation is in the form offz=fx+iy=ux,y+ivx,y , such thatux,y=xandvx,y=0 .

Hence, the real part of the given function is ux,y=x, and the imaginary part isvx,y=0 .

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Most popular questions from this chapter

Describe the Riemann surface for .w=z

Prove the theorem stated just after (10.2) as follows. Let φ(u, v) be a harmonic function (that is, φ satisfies ∂2φ/∂u2+∂2φ/∂v2=0). Show that there is then an analytic function g(w) = φ(u, v) + iψ(u, v) (see Section 2). Let w = f(z) = u + iv be another analytic function (this is the mapping function). Show that the function h(z) = g(f(z)) is analytic. Hint: Show that h(z) has a derivative. (How do you find the derivative of a function of a function, for example, ln sin z?) Then (by Section 2) the real part of h(z) is harmonic. Show that this real part is φ(u(x, y), v(x, y)).

Use the Cauchy-Riemann conditions to find out whether the functions in Problems 1.1 to 1.21 are analytic.

z¯

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

1z

Use the Cauchy-Riemann conditions to find out whether the functions in Problems 1.1 to 1.21 are analytic.

ez
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