Images under exp in complex plane

In summary, The conversation is discussing how to find the image of a line with the equation y = x under exp(z). The speaker suggests using parametric form and discusses the use of rectangular or polar coordinates. They also mention using euler's number in the complex plane. The conversation ends with a question about being able to write the image in parametric form. The limits as t approaches infinity and negative infinity are also mentioned.
  • #1
erraticimpulse
55
0
Describe the image under exp of the line with equation y = x. To do this you should find an equation (at least parametrically) for the image (you can start with the parametric form x = t; y = t), plot it reasonably carefully, and explain what happens in the limits as t approaches infinity and t approaches negative infinity.

Not very certain where I should start. Should I use rectangular coordinates or polar?
 
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  • #2
Here exp is notation for euler's number in the complex plane.
 
  • #3
You have the line in parametric form, are you able to write down it's image under exp(z) in parametric form?
 

1. What are images under exp in the complex plane?

Images under exp in the complex plane are points on the complex plane that result from applying the exponential function to a given point. This function maps a complex number to another complex number and is represented as z = exp(x + iy) = e^x(cos(y) + isin(y)), where x and y are real numbers.

2. How do images under exp behave in the complex plane?

Images under exp in the complex plane exhibit rotational and stretching behavior. The magnitude of the complex number increases as it moves away from the origin, while the angle of the number also changes.

3. Can images under exp be used to represent complex numbers?

Yes, images under exp can be used to represent complex numbers. The exponential function can be used to convert a complex number from rectangular form (a + bi) to polar form (r(cos(theta) + isin(theta))). This allows for easier calculation and visualization of complex numbers.

4. What are the applications of images under exp in the complex plane?

Images under exp have numerous applications in mathematics, physics, and engineering. They are used in solving differential equations, studying fractals, and understanding the behavior of waves and oscillations.

5. How is the concept of images under exp related to complex analysis?

The concept of images under exp is closely related to complex analysis, which is the study of complex-valued functions and their properties. The exponential function is an important tool in complex analysis and is used to define other important functions, such as the trigonometric functions and logarithmic functions.

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