Help with the next complex variable limit?

( #(Imz^2 )/"z+Re(z)"#
where #z->1-i#

1 Answer
Jul 12, 2018

Answer:

# = (- 4 - 2i)/5#

Explanation:

#f(z) = (bbb(Im)(z^2))/(z+bbb(Re)(z))#

#:. f(1 - i) = (- 2)/((2- i) )#

#= (- 2(2+i))/((2 - i)(2 + i)) = (- 4 - 2i)/5#

The function is:

#f(z) = ( 2 xy )/(2x+i y)#

# = (4 x^2y )/(4x^2+ y^2) -i ( 2xy^2)/(4x^2+ y^2) #

By inspection, the function is continuous at all points [except maybe the origin]. So the limit is simply the value of the function at the point in question, ie #f(1 - i) #.