Question

Suppose f is entire, with real and imaginary parts u and v satisfying u(x, y) v(x, y) = 3 for

all z = x + iy. Show that f is constant.

Be clearly, please. Do not upload same answers from others on Chegg. THANKS

Answer #1

Suppose,

f ( z) = u(x,y) + iv(x,y) is entire and u(x,y)v(x,y) =3 i.e. uv is constant

Consider (f(z))^{2} ,

As f(z) is entire
(f(z))^{2} is entire

and (f(z))^{2}= (u(x,y) + iv(x,y))^{2}

= (u(x,y))^{2} - (v(x,y))^{2 }+ 2i
u(x,y)v(x,y)

= (u(x,y))^{2} - (v(x,y))^{2} + 6i

As u(x,y) v(x,y) is Constant and (f(z))^{2} is
entire

therefore By CR equations ,

(u(x,y))^{2} - (v(x,y))2 is constant

(f(z))^{2}
is constant

Hence f(z) is constant.

Suppose f is entire, with real and imaginary parts u and v
satisfying u(x, y) v(x, y) = 3 for
all z = x + iy. Show that f is constant.

Please show all steps, thank you:
Problem C: Does there exist an analytic function f(z) in some
domain D with the real part u(x,y)=x^2+y^2?
Problem D: Is the function f(z)=(x-iy)^2 analytic in any domain
in C? Are the real part u(x,y) and the imaginary pary v(x,y)
harmonic in C? Are u and v harmonic conjugates of each other in any
domain?

The real part of a f (z) complex function is given as
(x,y)=y^3-3x^2y. Show the harmonic function u(x,y) and find the
expressions v(x,y) and f(z). Calculate f'(1+2i) and write x+iy
algebraically.

Let s = f(x; y; z) and x = x(u; v; w); y = y(u; v; w); z = z(u;
v; w). To calculate ∂s ∂u (u = 1, v = 2, w = 3), which of the
following pieces of information do you not need?
I. f(1, 2, 3) = 5
II. f(7, 8, 9) = 6
III. x(1, 2, 3) = 7
IV. y(1, 2, 3) = 8
V. z(1, 2, 3) = 9
VI. fx(1, 2, 3)...

Verify the Caucy-riemann equations for the functions u(x,y),
v(x,y) defined in the given domain
u(x,y)=x³-3xy², v(x,y)=3x²y-y³, (x,y)ɛR
u(x,y)=sinxcosy,v(x,y)=cosxsiny (x,y)ɛR
u(x,y)=x/(x²+y²), v(x,y)=-y/(x²+y²),(x²+y²), (
x²+y²)≠0
u(x,y)=1/2 log(x²+y²), v(x,y)=sin¯¹(y/√¯x²+y²), ( x˃0 )
In each case,state a complex functions whose real and imaginary
parts are u(x,y) and v(x,y)

Let f : R → R be a function satisfying |f(x) − f(y)| ≤ 3|x −
y|^{1/2} for all x, y ∈ R. Apply E − δ definition to show that f is
uniformly continuous in R.

Suppose you have the following preferences u(x,y) = v(x) + y.
Calculate the optimal demand functions. Is good x an
ordinary or giffen good? Please show
work.

Suppose f is a differentiable function of x
and y, and
g(u, v) =
f(eu
+ sin(v),
eu +
cos(v)).
Use the table of values to calculate
gu(0, 0)
and
gv(0, 0).
f
g
fx
fy
(0, 0)
0
5
1
4
(1, 2)
5
0
6
3
gu(0, 0)
=
gv(0, 0)
=

Evaluate the following.
f(x, y) = x + y
S: r(u, v) = 5
cos(u) i + 5 sin(u)
j + v k, 0 ≤ u
≤ π/2, 0 ≤ v ≤ 3

Write down the parametrized surfaces as level surfaces
{f(x,y,z)=0}.
x=ucosv, y=usinv, z=u, 0 <= u <= 2, 0 <= v <=
2pi
x = 2cosu*cosv, y = 2cosu*sinv, z = 2sinu, 0 <= u <= 2pi,
0 <= v <= pi

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 19 minutes ago

asked 22 minutes ago

asked 29 minutes ago

asked 37 minutes ago

asked 39 minutes ago

asked 47 minutes ago

asked 48 minutes ago

asked 48 minutes ago

asked 52 minutes ago

asked 52 minutes ago

asked 56 minutes ago