1

I have been studying Euler's Formula and its derivation. In an article I am reading, I came across a use of derivatives I did not understand and am hoping someone can explain it. The use of derivatives is in this article.

Let $z= a+bi$ for $e^z$ and $e^{a+bi}$

From the law of exponents, $e^{a+bi}= (e^a)(e^{bi})$

This where the confusion is

Derivative $\dfrac{d(e^{a+bi})}{da}= \dfrac{d(e^a)}{da} \dfrac{d (e^{bi})}{da}$

In the complex plane, $e^{bi}$ does not depend on $a$, and as $a$ changes $e^{bi}$ does not change at all, it is constant and thus $\frac{d(e^{bi})}{da} = 1.$ This is used to show that $\frac{d(e^a)}{da}1= e^a.$ Should not the derivative of a constant term be zero.

copper.hat
  • 172,524
Elliott
  • 65
  • 3
    Why do you think $d(e^{a+bi})/da= d(e^a)/da * d(e^{bi})/da$? Are you trying to use the product rule? Also, the link appears to be dead. – Braindead Dec 15 '14 at 19:53
  • 1
    I see nothing whatsoever about derivatives in the linked-to article. – Santiago Canez Dec 15 '14 at 20:12
  • The link to the article does appear to be working. The article is www.mathed.soe.vt.edu/Undergraduates/Euler Explanation.pdef. The use of the said derivative relationship is discussed about 5 pages into the article and it makes no sense to me. I do t understand the approach used in the article. elliott – Elliott Dec 15 '14 at 20:52
  • @Elliott I don't think the new link in the body of your post links to the correct article. The link provided by bourbaki works. Now, I didn't read the entire article, but that specific part with the usage of the derivative is completely wrong, which is why you might be confused by it. – Braindead Dec 19 '14 at 01:09

1 Answers1

1

By the way, the correct link seems to be http://www.mathed.soe.vt.edu/Undergraduates/EulersIdentity/EulersExplanation.pdf

But I don't think it makes any sense. For example, $$\frac {\text d}{\text d a} e^{a+bi}\neq \frac {\text d}{\text d a} e^{a} \frac {\text d}{\text d a} e^{bi}$$

Looking at the definition of $e^z$: $$e^z = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \dots = \sum_{n=0}^{\infty} \frac{z^n}{n!} $$

we can see that $$\frac {\text d}{\text d a} e^{a+bi} = e^{a+bi}$$

but I don't get the point of this paper as it doesn't really explain that much about anything except what polar coordinates are...

neptun
  • 1,573