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 $\frac{\theta^2}{2!}$ element - on figure [9] we can see $\frac{i\theta^2}{2!}$? Likewise why it contains $\frac{\theta^5}{5!}$ 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 $\frac{\theta^2}{2!}$ element - on figure [9] we can see $\frac{i\theta^2}{2!}$? Likewise why it contains $\frac{\theta^5}{5!}$ while on fig [9] we see this is a real number?


Thanks

$S(\theta)$ only contains the imaginary parts of the series and $C(\theta)$ only the real parts.

 

[Mondo]

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

 

[blamocur]

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

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

 

[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 $\frac{\theta^3}{3!}$ 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 $\frac{\theta^3}{3!}$ 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. $e^\theta$, while the complex case $e^{i\theta}$ 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 $\frac{\theta^2}{2!}$ element - on figure [9] we can see $\frac{i\theta^2}{2!}$? Likewise why it contains $\frac{\theta^5}{5!}$ while on fig [9] we see this is a real number?


Thanks

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

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

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

 

[Mondo]

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