Complex analysis and delta epsilon proof

[buzzmath]

Can anyone help with these problems

1). use def. delta epsilon proof to prove lim(z goes to z0) Re(z)=Re(z0)

This is what I did |Re(z)-Re(z0)| = |x-x0| < epsilon then |z-z0|=|x-iy-x0-iy0|=|x-x0+i(y-y0)|<=|x-x0|+|y-y0|=epsilon + |y-y0| = delta
My question is doesn't this delta have to be a function of epsilon so is it alright to write it with this |y-y0| value in there? I don't see how to do it otherwise

2). lim(z goes to z0) conjugate of z = conjugate of z0

the way I see this problem as since the conjugate is just a reflection over the real axis then the delta value would be the same as the epsilon value because the points are the same distance away |z-z0|=|conjugate(z)-conjugate(z0)|
but how exactly would you go about writing this as a formal proof?

3). One interpretation of a function w=f(z)=u(x,y)+iv(x,y) is that of a vector field in the domain of definition of f. The function assigns a vector w, with components u(x,y) and v(x,y), to each point z at which it is defined. Indicate graphically the vector fields represented by
a) w=iz
b). w=z/|z|

for a) I have u(x,y) = -y and v(x,y)=x and for b) I have
u(x,y)=x/sqrt(x^2 + y^2) and v(x,y)=y/sqrt(x^2 + y^2)
my question is how do you show graphically this vector field which I think is just the image without a domain?

thanks for any help

 

[mighty2000]

1). To prove the limit using the delta-epsilon definition, you need to show that for every epsilon > 0, there exists a delta > 0 such that whenever |z - z0| < delta, then |Re(z) - Re(z0)| < epsilon. In your proof, you have correctly chosen delta = epsilon + |y-y0|. This delta does not have to be a function of epsilon, it just needs to satisfy the condition for any epsilon. So it is fine to write it with |y-y0| in there.

2). To prove this limit, you can use a similar approach as in the first problem. For any epsilon > 0, you need to show that there exists a delta > 0 such that whenever |z - z0| < delta, then |conjugate(z) - conjugate(z0)| < epsilon. Since the conjugate of a complex number is just a reflection over the real axis, you can choose delta = epsilon and this will satisfy the condition. This can be written as a formal proof by stating the definition of the limit and then showing the steps to reach the conclusion.

3). To graphically represent the vector field, you can plot the vectors at different points in the domain of definition. For example, for w = iz, you can plot the vectors (0,1), (-1,0), (0,-1), (1,0) at different points in the domain. Similarly, for w = z/|z|, you can plot the vectors (1,0) and (0,1) at different points in the domain. This will give you a visual representation of the vector field. Keep in mind that the domain of definition for these functions will depend on the specific problem, so make sure to take that into account when plotting the vectors.