Ask Your Question
1

i exponents of non integers

asked 2022-05-09 08:38:14 +0200

Purrastroika gravatar image

I'm not very good at math, but always had a creative interest. Especially when it came to imaginary numbers. The question that occurred to me is How do you graph Y=i multiplied by itself to the power of a non-whole number (I think the word is "non-integer?")? Would that be an indefinite "tube," for lack of better verbage, between -1 and 1? Tangenting off the "tube" remark, would it affect the Z or other different axis? I didn't make it past precalculus, and I'm not in school. Just a random curiosity, and I'm not sure if it has any real application. Thank you in advance for any time and attention you spend replying.

edit retag flag offensive close merge delete

Comments

Trying :

plot([(I^x).abs(), (I^x).maxima_methods().arg()], (x, -3, 3), legend_label=["Modulus", "Argument"])

may enlighten you.

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2022-05-09 15:16:10 +0200 )edit

1 Answer

Sort by ยป oldest newest most voted
1

answered 2022-05-09 16:00:15 +0200

Emmanuel Charpentier gravatar image

updated 2022-05-09 16:05:23 +0200

As suggested above, a graphical representation gives an easy intuition :

plot([(I^x).abs(), (I^x).maxima_methods().arg()], (x, -3, 3), legend_label=["Modulus", "Argument"])

Argument an modulus of $e^{ix}$

To understand the result, try :

sage: var("a, b")
(a, b)
sage: E0=log(a^b)==log(a^b).log_expand() ; E0
log(a^b) == b*log(a)
sage: E1 =E0.subs([a==I, b==x]) ; E1
log(I^x) == 1/2*I*pi*x
sage: E2 = E1.operator()(*map(exp, E1.operands())) ; E2
I^x == e^(1/2*I*pi*x)

The latter may be easier to grasp in a different form :

sage: E2.rhs().demoivre(force=True)
cos(1/2*pi*x) + I*sin(1/2*pi*x)

Exercise for the (advanced) reader : try and understand :

complex_plot(I^x, (-3, 3), (-3, 3))

Complex plot of $e^{iz}$

Hint : look up the definitions of exp and trig functions for complexes...

HTH,

edit flag offensive delete link more

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

Stats

Asked: 2022-05-09 08:38:14 +0200

Seen: 22 times

Last updated: May 09