- #1

NewtonianAlch

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## Homework Statement

Determine where the function f(x + iy) = 2sin(x) + iy^2 + 4(ix - y) is differentiable and where it is analytic.

## The Attempt at a Solution

f(x + iy) = 2sin(x) -4y + i(y^2 +4x)

Through C-R equations:

du/dx = 2 cos x

dv/dy = 2y

du/dy = -4

dv/dx = 4

So the C-R equations hold only if y = cos (x)

1) Supposedly these partial derivatives are continuous everywhere, I do not understand what this exactly means, what is an example of a partial derivative that is not continuous everywhere?

Also it says f is differentiable only on the curve y = cos (x), but it's not analytic since it's not differentiable in the e-neighbourhood at a point on the curve.

2) I do know that for a function to be analytic it must be differentiable at a given point,

**and**also in the e-neighbourhood of it, but how is this exactly determined, and what does it mean to be differentiable in an e-neighbourhood of a point in this curve specifically?

If I draw out y = cos (x), pick out a point and look at the e-neighbourhood around this point, all you get is the point on that line, a portion of the line going through, and nothing else around it.