Composing Two Complex Functions

[Bashyboy]

Homework Statement

Suppose we have the function $f(z) = x + iy^2$ and a contour given by $z(t) = e^t + it$ on $a \le t \le b$.
Find $x(t)$, $y(t)$, and $f(z(t))$.

Homework Equations

The Attempt at a Solution

Well, $x(t)$ and $y(t)$ are rather simple to identity. However, I am having difficulty determining $f(z(t))$, which I believe seems from some notational issues.

$f(z) = f(z(t)) = f(\underbrace{e^t + it}_?) = ...?$

I could write

$f(z) = x + iy^2 \iff$

$f\langle (x,y) \rangle = x + iy^2$.

I know that $x(t) = e^t$ and $y(t) = t$.

$f\langle (x(t), y(t) ) \rangle = x(t) + i[y(t)]^2 \iff$

$f\langle (x(t), y(t) ) \rangle = e^t + it^2$.

I find this somewhat unsettling. Suppose that I have the function $f(z) = f(x,y) = u(x,y) + iv(x,y)$ (I am dropping the $\langle \rangle$ notation); and suppose that we have the contour described by the parametric function $z(t) = x(t) + iy(t)$. In the general case, would the composition look like

$f(x(t),y(t)) = u(x(t),y(t)) + iv(x(t),y(t))$?

 

[Mark44]

Homework Statement

Suppose we have the function $f(z) = x + iy^2$ and a contour given by $z(t) = e^t + it$ on $a \le t \le b$.
Find $x(t)$, $y(t)$, and $f(z(t))$.

Homework Equations

The Attempt at a Solution

Well, $x(t)$ and $y(t)$ are rather simple to identity. However, I am having difficulty determining $f(z(t))$, which I believe seems from some notational issues.

$f(z) = f(z(t)) = f(\underbrace{e^t + it}_?) = ...?$

I could write

$f(z) = x + iy^2 \iff$

$f\langle (x,y) \rangle = x + iy^2$.

I know that $x(t) = e^t$ and $y(t) = t$.

$f\langle (x(t), y(t) ) \rangle = x(t) + i[y(t)]^2 \iff$

$f\langle (x(t), y(t) ) \rangle = e^t + it^2$.

Looks fine to me. f maps a complex number x + iy to x + iy², so the same function maps e^t + it to e^t + it², which is what you have.

I find this somewhat unsettling. Suppose that I have the function $f(z) = f(x,y) = u(x,y) + iv(x,y)$ (I am dropping the $\langle \rangle$ notation); and suppose that we have the contour described by the parametric function $z(t) = x(t) + iy(t)$. In the general case, would the composition look like

$f(x(t),y(t)) = u(x(t),y(t)) + iv(x(t),y(t))$?