Question

Can you find a functionv (x,y) so that u+iv is entire with u(x,y) =x^3+ 3xy^2 ? (cauchy- riemann equations---hint)

Answer #1

Two functions, u(x,y) and v(x,y), are said to verify the
Cauchy-Riemann
differentiation equations if they satisfy the following
equations ∂u\dx=∂v/dy and ∂u/dy=−(∂v/dx)
a. Verify that the Cauchy-Riemann differentiation equations can
be written in the polar coordinate form as
∂u/dr=1/dr ∂v/dθ and ∂v/dr =−1/r ∂u/∂θ
b. Show that the following functions satisfy the Cauchy-Riemann
differen- tiation equations
u=ln sqrt(x^(2)+y^(2)) and v= arctan y/x.

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)

Solve the following nonhomogenous Cauchy-Euler equations for x
> 0.
a. x^(2)y′′+3xy′−3y=3x^(2).

Find the equation of the tangent line
x^2 – 3xy^2 + y^3 = -3 at (2, -1)

f(x, y) = 4 + x^3 + y^3 − 3xy
(a,b)=(0.5,0.5)
u = ( √ 1 /2 , − √ 1 /2 )
a) Calculate the rate of change of f along the curve r(t) = (t,
t2 ), at t = −1
b)Classify the critical points of f using the second derivative
test.

Find dy/dx by implicit differentiation:
A.). x^4 + y^3 = 3
B.). 5x^2 +3xy - y^2 =7
C.) x^3(x+y) = y^2(4x-y)

Solve the IVP with Cauchy-Euler ODE: x^2 y''+3xy'+4y=0; y(1)=0,
y’(1)=−2

x^2y'' − 3xy'+ 4y = 0 ; y(1)=5 y'(1)=3
differential equation using the Cauchy-Euler method

x^2y'' − 3xy'+ 4y = 0 ; y(1)=5 y'(1)=3
differential equation using the Cauchy-Euler method

Find the area enclosed by the loop of the curve x^3 + y^3
=3xy

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 10 minutes ago

asked 13 minutes ago

asked 27 minutes ago

asked 27 minutes ago

asked 28 minutes ago

asked 34 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago