Integer Raised To Complex Number Power
I cannot find method, or figure one out, to raise a non-zero integer to a complex number power. Example 1: 2 raised to the 2 +3i power. Example 2: -3 raised to the 2 + 3i power. Example 3: 1/4 with the 4 raised to the 3 + 4i power. Can someone enlighten me as to how this is done, preferable in a general manner that can be applied to all problems of this nature.
Not What Looking For, Gusbob
Gusbob -- Thank you for replying to my question. However, you have not answered it. Wolfram MathWorld section titled "Complex Exponentiation" starts off by saying, "A complex number may be taken to the power of another complex number. . . ."
1. This is not what I am asking. Rather, it is: If you have a not-zero integer (1, 2, 3,. . . or -1, -2, -3,. . .), how do you raise it to a complex power. If I want to raise the postive integer 2 to the complex number power 2 + 2i, how do I do it?
2 Also, I realize that I am also asking that if you have a non-zero rational number (i.e., a fraction) how do you raise the divisor to a complex power. This is basically the same as No. 1 above.
Thank you, Plato (I Think)
Plato: I would do a MHF "official" thank you, but I am a "Newbie" and, as yet don't know how to do this.
Otherwise: Yikes! Back to the mental "drawing board" for me! Do you know of a Website (and location therein) where I might find more detailed information on my question? And, could you recommend a book or textbook I might purchase from, say, Amazon.Com that would help me -- in, frankly, what seems to be "an impossible quest"
You no doubt have surmised -- I am a "Newbie" with regards to Mathematics as well as MHF. But, since I was about 10 years old I have always had a deep and sincere interest, but almost totally unpursed, in Mathematics. My question -- the first here in MHF -- related to my (no doubt what will be very long term) desire to understand the Reimann Hypothesis.
Any help you (or other members of this forum) can provide deeply appreciated (and very much needed).
Plato (Or Others) -- Please Clarify Post
Plato (Or Others) --
1. Is the "e" you refer to the natural base e, the irrational number equal to approx. 2.71828?
2. If it is, when you say, "You need to know how to raise e to a complex number.," what do you mean? That is, how do you "raise" an irrational number to a complex power? I am really confused!
3. From your two examples it seems that to answer my question (to myself) I need to thoroughly know and understand --
b. Polar Coordinates.
c. DeMoivre's Theorem.
Regarding 3 above, is "a," "b," and "c" a correct assumption on my part?
4. Is there anything else additional I need to know before I am equipped to raise an integer or a real number to a complex power?