How to rebuild the analytical function f(z) if given #U(x,y)=Re f(z)=2x+(x/(x^(2)+y^(2)))# #0<|z|<+oo#?
2 Answers
# f(z) = (2x+x/(x^2+y^2)) + 1i #
Explanation:
By the very definition of
Suppose that we define:
# f(z) = U(x,y) + V(x,y)i #
# \ \ \ \ \ \ \ = (2x+x/(x^2+y^2)) + V(x,y)i #
Where
Then we require:
# |U^2(x,y)+V^2(x,y)| > 0 #
# :. |(2x+x/(x^2+y^2)) + V(x,y)i| > 0 #
# :. sqrt((2x+x/(x^2+y^2))^2 + V^2) > 0 #
# :. (2x+x/(x^2+y^2))^2 + V^2 > 0 #
Clearly:
# 0 le (2x+x/(x^2+y^2))^2 < oo #
So we can choose any function
Thus one such function is:
# f(z) = (2x+x/(x^2+y^2)) + 1i #
See below.
Explanation:
If
These equations occurred already in the 18th century in J.L. d'Alembert's and L. Euler's studies on functions of a complex variable.
Here
but
and also
but
then
so finally
Concluding,