Show that this function is analytic

[stunner5000pt]

Show that this function is analytic

$$\left( x + \frac{x}{x^2 + y^2} \right) + i \left( y - \frac{y}{x^2 + y^2} \right)$$

now... would i substitute $$x = \frac{z + \overline{z}}{2}$$
and
$$y = \frac{z - \overline{z}}{2}$$

and then see if z or z bar appear exlicitly in the function??
Would that solve it??

Is there an easier way? A less Messy way?

 

[TD]

If I recall correctly, a function is analytic if it is differentiable and if it satisfies the Cauchy-Riemann equations, perhaps you should check those?

 

[stunner5000pt]

is there another way of doing it... perhaps using the substitutions i suggested above and then checking the domain of the function?

 

[d_leet]

is there another way of doing it... perhaps using the substitutions i suggested above and then checking the domain of the function?

Using the Cauchy-Riemann equations is probably the easier way.

 

[d_leet]

Show that this function is analytic

$$\left( x + \frac{x}{x^2 + y^2} \right) + i \left( y - \frac{y}{x^2 + y^2} \right)$$

now... would i substitute $$x = \frac{z + \overline{z}}{2}$$
and
$$y = \frac{z - \overline{z}}{2}$$

and then see if z or z bar appear exlicitly in the function??
Would that solve it??

Is there an easier way? A less Messy way?

Your equation for y should be over 2i not just 2.

 

[TD]

is there another way of doing it... perhaps using the substitutions i suggested above and then checking the domain of the function?

There are complex functions whose domain is C but are nowhere analytic.