Complex powers/logarithmic spirals

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In summary, adding any integer multiple of (2*pi*i) to log w is equivalent to a full rotation in the complex plane. This step is equivalent to multiplying w^z by e^(z*2*pi*i). This is represented in the complex plane as the intersections of 2 logarithmic spirals, with the first spiral given by w^z. The other spiral is given by e^(z(log w + 2*pi*i)). This can be understood using rules of exponents from algebra. These concepts are further explained in pages 96-97 of The Road to Reality by Roger Penrose.
  • #1
hoodwink
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When looking at

w^z = e^(z log w)

I understand that adding any integer multiple of (2*pi*i) to log w is equivalent to a full rotation in the complex plane. I don't understand how this step is equivalent to multiplying w^z by e^(z*2*pi*i). Also, I'm missing the significance of this being represented in the complex plane as the intersections of 2 logarithmic spirals. I can see how the first spiral is given by w^z, but the other?

If anyone has a copy handy, my questions arose from looking at pages 96-97 of The Road to Reality by Roger Penrose.
 
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  • #2
I think it is just rules of exponents from algebra.

(w^z)*e^(z*2*pi*i) = e^(z log w)*e^(z*2*pi*i) = e^(z log w + z*2*pi*i) =
e^(z(log w + 2*pi*i))

Does this help?
 
  • #3
click! thanks for that diffy. hopefully those spirals will start to do the same now...:smile:
 

1. What are complex powers?

Complex powers are a type of mathematical function where a complex number is raised to another complex number. This results in a complex number as the answer.

2. How can complex powers be represented graphically?

Complex powers can be represented graphically as a logarithmic spiral. This is a spiral curve that increases or decreases in size as it moves away from the origin, with the rate of increase or decrease determined by the complex number.

3. What is a logarithmic spiral?

A logarithmic spiral is a type of spiral curve in which the distance from the center to any point on the spiral is proportional to the logarithm of the polar angle of that point from the origin.

4. How are logarithmic spirals used in real life?

Logarithmic spirals have many applications in real life, including in architecture, art, and nature. They can be seen in the shape of a snail's shell, hurricane formations, and the arrangement of seeds in a sunflower.

5. What is the connection between complex powers and logarithmic spirals?

The connection between complex powers and logarithmic spirals is that complex powers can be represented graphically as logarithmic spirals. The complex number used in the power determines the rate of increase or decrease of the spiral, and the angle of rotation determines the direction of the spiral.

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