91Ó°ÊÓ

A parametric curve in complex analysis is a curve that is defined using a parameter. In our exercise, the parametric curve is given by the equation \(z(t) = i + e^{it}\), where \(t\) is the parameter that varies over the interval \([0, \pi]\). This parameter \(t\) controls how the point \(z(t)\) moves along the curve in the complex plane.
To better visualize this, we can rewrite \(e^{it}\) using its trigonometric form: \(e^{it} = \cos(t) + i\sin(t)\). Hence, the curve becomes \(z(t) = \cos(t) + (1+\sin(t))i\).
This means the real part of \(z(t)\) is \(\cos(t)\) and the imaginary part is \(1 + \sin(t)\), forming a semicircle. In essence, the parametric curve here represents a movement from the point \((0, 1)\) at \(t = 0\), tracing a semicircle, and ending back below at \(t = \pi\) in the complex plane.

Complex mapping involves transforming a complex function \(z\) into another complex function \(w = f(z)\). In our exercise, the mapping given is \(f(z) = (z-i)^3\). This function transforms our initial parametric curve \(C\) into a new curve \(C'\) in the complex plane.
To understand this, we replace \(z\) with our parametric equation \(z(t) = i + e^{it}\), then under the mapping \(w = f(z) = (z-i)^3\), substitute to get \(w(t) = (e^{it})^3 = e^{3it}\).
Through this mapping, the initial curve transforms by scaling and rotating according to the formula. It effectively stretches the angle threefold, compressing the semicircle into a shorter angular distance. Thus, complex mapping can drastically change the shape and size of the curve being transformed, as seen with the effect on curve \(C'\).

The complex plane is a two-dimensional plane for representing complex numbers graphically. It is often depicted where the horizontal axis (x-axis) represents the real part of the complex number and the vertical axis (y-axis) represents the imaginary part.
For the curve \(z(t) = i + e^{it}\), we interpret its behavior on this plane. The curve begins at \(1i\) (purely imaginary, since the cosine term vanishes at \(t = 0\)) and moves as \(t\) increases, drawing a semicircle.
The image curve \(C'\) described by \(w(t) = e^{3it}\) is similarly plotted in the complex plane, illustrating how the mapping \((z-i)^3\) changes the shape. This underscores how the complex plane is a crucial tool in visualizing the transformations complex numbers undergo.

The exponential form of a complex number is a powerful way to express the dynamics within complex analysis. The original parametric form \(z(t) = i + e^{it}\) uses this form with \(e^{it} = \cos(t) + i\sin(t)\), illustrating a clean and simple representation of the trigonometric components.
This form is instrumental in revealing the circular nature of the movement as \(e^{it}\) traces a unit circle in the complex plane. Additionally, the mapping ensures that the function could be expressed as \((e^{it})^3 = e^{3it}\).
Using exponential form aids in simplifying expressions, calculating powers of complex numbers easily, and understanding rotational symmetries, making it indispensable in both theoretical and applied complex analysis.