ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 09 May 2022 16:00:15 +0200i exponents of non integershttps://ask.sagemath.org/question/62367/i-exponents-of-non-integers/ 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.
Mon, 09 May 2022 08:38:14 +0200https://ask.sagemath.org/question/62367/i-exponents-of-non-integers/Comment by Emmanuel Charpentier for <p>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.</p>
https://ask.sagemath.org/question/62367/i-exponents-of-non-integers/?comment=62370#post-id-62370Trying :
plot([(I^x).abs(), (I^x).maxima_methods().arg()], (x, -3, 3), legend_label=["Modulus", "Argument"])
may enlighten you.Mon, 09 May 2022 15:16:10 +0200https://ask.sagemath.org/question/62367/i-exponents-of-non-integers/?comment=62370#post-id-62370Answer by Emmanuel Charpentier for <p>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.</p>
https://ask.sagemath.org/question/62367/i-exponents-of-non-integers/?answer=62371#post-id-62371As 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 $e^{ix}$](/upfiles/16521044095883759.png)
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 $e^{iz}$](/upfiles/16521046104800829.png)
Hint : look up the definitions of `exp` and trig functions for complexes...
HTH,Mon, 09 May 2022 16:00:15 +0200https://ask.sagemath.org/question/62367/i-exponents-of-non-integers/?answer=62371#post-id-62371