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 | 1 | initial version |
The limit is equivalent to two limits of functions of two real variables
x,y=var('x y',domain=RR)
z=x+I*y
f=(z.conjugate()/z)^2
f, f.rectform()
((x - I*y)^2/(x + I*y)^2,
-4*I*(x^2 - y^2)*x*y/(x^2 + y^2)^2 - (4*x^2*y^2 - (x^2 - y^2)^2)/(x^2 + y^2)^2)
It does not exist, since the limits on some subsets are different
limit(f.subs(y=0),x=0), limit(f.subs(x=y),y=0)
(1, -1)
Use
plot3d(real_part(f),(x,-1,1),(y,-1,1))
plot3d(imag_part(f),(x,-1,1),(y,-1,1))
for better understanding
| | 2 | No.2 Revision |
The limit is equivalent to two limits of functions of two real variables
x,y=var('x y',domain=RR)
z=x+I*y
f=(z.conjugate()/z)^2
f, f.rectform()
((x - I*y)^2/(x + I*y)^2,
-4*I*(x^2 - y^2)*x*y/(x^2 + y^2)^2 - (4*x^2*y^2 - (x^2 - y^2)^2)/(x^2 + y^2)^2)
It does not exist, since the limits on some subsets are different
limit(f.subs(y=0),x=0), limit(f.subs(x=y),y=0)
(1, -1)
Use
plot3d(real_part(f),(x,-1,1),(y,-1,1))
plot3d(imag_part(f),(x,-1,1),(y,-1,1))
for better understanding
Using another approach, if a is the modulus and b the argument, then
a,b=var('a b',domain=RR)
z=a*exp(I*b)
f=(z.conjugate()/z)^2;f
e^(-4*I*b)
and the limit
limit(f,a=0)
e^(-4*I*b)
depends on the argument , so the limit with z->0 does not exist
