Euler formula derived from a power series

[Mondo]

Hello,

I am reading Visual complex analysis book and got stuck at page 12 where the author derives Euler formula from a power series as shown below:


Basically figure [9] shows that real numbers are horizontal while imaginary ones are 90 degrees apart. However I don't understand how the author derives C(0) and S(0). For example, why S(0) doesn't contain [imath]\frac{\theta^2}{2!}[/imath] element - on figure [9] we can see [imath]\frac{i\theta^2}{2!}[/imath]? Likewise why it contains [imath]\frac{\theta^5}{5!}[/imath] while on fig [9] we see this is a real number?


Thanks

 

[blamocur]

Hello,

I am reading Visual complex analysis book and got stuck at page 12 where the author derives Euler formula from a power series as shown below:
View attachment 30431

Basically figure [9] shows that real numbers are horizontal while imaginary ones are 90 degrees apart. However I don't understand how the author derives C(0) and S(0). For example, why S(0) doesn't contain [imath]\frac{\theta^2}{2!}[/imath] element - on figure [9] we can see [imath]\frac{i\theta^2}{2!}[/imath]? Likewise why it contains [imath]\frac{\theta^5}{5!}[/imath] while on fig [9] we see this is a real number?


Thanks

[imath]S(\theta)[/imath] only contains the imaginary parts of the series and [imath]C(\theta)[/imath] only the real parts.

 

[Mondo]

@blamocur yes but this does not answer my question why? Why is [imath]\frac{\theta^3}{3!}[/imath] considered imaginary and not real?

 

[blamocur]

@blamocur yes but this does not answer my question why? Why is [imath]\frac{\theta^3}{3!}[/imath] considered imaginary and not real?

Because it is actually [imath]\frac{(i\theta)^3}{3!}[/imath], which is equal to [imath]-i\frac{\theta^3}{3!}[/imath]. Remember, in the Euler formula the sine comes multiplied by [imath]i[/imath], i.e. [imath]e^{i\theta} = \cos\theta + \mathbf i \sin \theta[/imath].

 

[Mondo]

Ok but then the whole figure [9] is nothing but confusion - in the description author says horizontal numbers are all real and then he make [imath]\frac{\theta^3}{3!}[/imath] an imaginary part. I also don't believe this paragraph is even close to derive Euler formula.

You are justifying this by using your own experience/knowledge not based on the page I pasted.

 

[blamocur]

Ok but then the whole figure [9] is nothing but confusion - in the description author says horizontal numbers are all real and then he make [imath]\frac{\theta^3}{3!}[/imath] an imaginary part. I also don't believe this paragraph is even close to derive Euler formula.

You are justifying this by using your own experience/knowledge not based on the page I pasted.

You are right: I've missed that detail. The horizontal line illustrates the purely real case, i.e. [imath]e^\theta[/imath], while the complex case [imath]e^{i\theta}[/imath] is illustrated by the rectangular spiral.
Sorry for the confusion.

 

[Dr.Peterson]

Hello,

I am reading Visual complex analysis book and got stuck at page 12 where the author derives Euler formula from a power series as shown below:
View attachment 30431

Basically figure [9] shows that real numbers are horizontal while imaginary ones are 90 degrees apart. However I don't understand how the author derives C(0) and S(0). For example, why S(0) doesn't contain [imath]\frac{\theta^2}{2!}[/imath] element - on figure [9] we can see [imath]\frac{i\theta^2}{2!}[/imath]? Likewise why it contains [imath]\frac{\theta^5}{5!}[/imath] while on fig [9] we see this is a real number?


Thanks

First, the picture doesn't show [imath]\frac{i\theta^2}{2!}[/imath]; it shows [imath]\frac{(i\theta)^2}{2!}[/imath], which is a negative real number. Is that what you meant?

Second, [imath]S(\theta)[/imath] is a real number, which is the imaginary part of [imath]e^{i\theta}[/imath].

Did you write out the expansion of [imath]e^{i\theta}[/imath], and see that alternating terms are imaginary? Please do so, and show your work. You can't see where [imath]C(\theta)[/imath] and [imath]S(\theta)[/imath] come from without doing that.

 

[Mondo]

Ok I think I see it now - every even power in the power series expansion of [imath]e^{i\theta}[/imath] is a real number while the odd power is the imaginary counterpart.