How to tell that 1/(z+2i) is not analytic?

[nutcase21]

hi as shown above, i have come across this expression. But i am not sure how to tell if it is not analytic in the circle |z-2|=4, clockwise.
expression: 1/(z+2i)

On top of that, can anyone give me a general idea of how to see if the expression is analytic or not analytic without using the cauchy's equation.

pls help. thx.

 

[HallsofIvy]

Well, that circle contains the point -2i at which f(x) does not even exist.

 

[nutcase21]

but how do u tell that f(x) does not exist there?

does it mean the function will be come infinite when z=-2i and thus telling us it is not analytic at that point?

 

[chiro]

but how do u tell that f(x) does not exist there?

does it mean the function will be come infinite when z=-2i and thus telling us it is not analytic at that point?

Consider the properties of divergence and also other analytic properties like limits: the limit must be the same from both sides and continuity must hold (among the other properties of analytic functions).

 

[CellCoree]

To determine if 1/(z+2i) is analytic, we can use the Cauchy-Riemann equations. These equations state that a complex function is analytic if and only if it satisfies the Cauchy-Riemann equations, which are a set of necessary and sufficient conditions for a function to be complex differentiable at a point. In other words, if the partial derivatives of the real and imaginary parts of the function exist and are continuous, and satisfy the Cauchy-Riemann equations, then the function is analytic.

In this case, the function 1/(z+2i) can be written as f(z) = u(x,y) + iv(x,y), where u(x,y) = 1/(x^2+(y+2)^2) and v(x,y) = 1/(x^2+(y+2)^2). By taking the partial derivatives of u and v with respect to x and y, we can see that u_x = -2x/(x^2+(y+2)^2)^2, u_y = -2(y+2)/(x^2+(y+2)^2)^2, v_x = 2x/(x^2+(y+2)^2)^2, and v_y = 2(y+2)/(x^2+(y+2)^2)^2.

Now, we can plug these into the Cauchy-Riemann equations: u_x = v_y and u_y = -v_x. However, when we try to solve for x and y, we end up with a contradiction. This means that the Cauchy-Riemann equations are not satisfied, and therefore the function 1/(z+2i) is not analytic.

To determine if a function is analytic without using the Cauchy-Riemann equations, we can also use the definition of analyticity. A complex function f(z) is analytic at a point z0 if and only if it has a power series representation in a neighborhood of z0. In other words, if we can write f(z) = ∑(n=0 to ∞) c_n(z-z0)^n, where c_n are complex constants, then the function is analytic at z0.

In this case, the function 1/(z+2i) does not have a power series representation in a neighborhood of z0 = -2i. Therefore, it is