Find the image of the infinite strip "-1<=x<=1" under the mapping w=e^z

General
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find the image of the infinite strip “-1<=x<=1” under the mapping w=e^z

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Finding the Image of the Infinite Strip -1 \leq x \leq 1 Under the Mapping w = e^z

Answer: To find the image of the infinite strip -1 \leq x \leq 1 under the exponential mapping w = e^z, we need to analyze how complex functions transform regions in the complex plane. The variable z is a complex number represented as z = x + iy, where x is the real part and y is the imaginary part. Therefore, the strip is defined by z = x + iy with -1 \leq x \leq 1. The function we are applying is w = e^z.

Step-by-Step Transformation Analysis

1. Basic Properties of the Exponential Function

The exponential function for complex numbers, w = e^z, can be expressed in terms of its real and imaginary components:

w = e^{x + iy} = e^x \cdot e^{iy}.

Using Euler’s formula, e^{iy} = \cos(y) + i\sin(y), we can rewrite this as:

w = e^x (\cos(y) + i\sin(y)).

This implies that:

  • The modulus (magnitude) of w is |w| = e^x.
  • The argument (angle) of w is \arg(w) = y.

2. Transformation of the Strip -1 \leq x \leq 1

Since -1 \leq x \leq 1, the modulus |w| will range from e^{-1} to e^1.

  • For x = -1: The modulus |w| at x = -1 is e^{-1} \approx 0.3679.
  • For x = 1: The modulus |w| at x = 1 is e^{1} \approx 2.7183.

3. Effect of the Imaginary Part y

The imaginary part y of z becomes the argument of the complex number w. Hence, as y varies from -\infty to \infty, it results in a complete traversal around the unit circle, creating a full circle in the imaginary exponential part.

Final Image Description

The image of the strip -1 \leq x \leq 1 under the transformation w = e^z is the annular region (ring-shaped) in the complex plane. It is bounded by two circles:

  • An inner circle with a radius r_1 = e^{-1}.
  • An outer circle with a radius r_2 = e.

As y varies, each value of x forms a circle centered at the origin, connecting to form an entire annular region between these two radii.

Geometric Interpretation

  • The complex exponential mapping converts vertical strips in the z-plane into annular regions in the w-plane.
  • The line segments formed by -1 \leq x \leq 1 correspond to circles at radial distances e^{-1} and e from the origin in the w-plane.
  • The width of the strip in the x direction directly influences the range of radii in the w-plane.

Summary

The transformation of the infinite strip -1 \leq x \leq 1 under the exponential mapping w = e^z results in an annular region centered at the origin in the w-plane. This annulus extends from radius r_1 = e^{-1} to r_2 = e, with y sweeping over all angles from 0 to 2\pi, thus substantially covering a full loop for each x value between the strip bounds.

If you have any more questions or need further clarification, feel free to ask! @anonymous4