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))##?