Math Help - Integer Raised To Complex Number Power

1. 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.

2. 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.

3. Originally Posted by DeanSchlarbaum
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.
Do you realise that real numbers (which include integers) are a subset of complex numbers ....?

4. Originally Posted by DeanSchlarbaum
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.
$2^{2 + 2i} =2^2 2^{2i} = 4(2^2)^i=4(4)^i$

How far do you want to evaluate this?

5. Originally Posted by mr fantastic
Do you realise that real numbers (which include integers) are a subset of complex numbers ....?
Yes, Gusbob, it is my understanding that Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, and Imaginary Numbers are all subsets of Complex Numbers, and can be expressed as Complex Numbers.

a + bi where a and b are real numbers and i is square root of -1

If b = 0, then a + bi is a real number

If a = 0, then a + bi is an imaginary number

The precalculus textbook I have gives general methods for adding, subtracting, multiplying, dividing complex numbers, and how to find the roots of negaive numbers.

What I am looking for is the General Method for raising a non-zero integer to a complex number power. Is there such a thing? Or, can you refer me to a book I might purchase (say at Amazon.com) that can, in fact, provide me with a General Method for doing this?

6. Originally Posted by AllanCuz
$2^{2 + 2i} =2^2 2^{2i} = 4(2^2)^i=4(4)^i$

How far do you want to evaluate this?
Allancuz, thank you for providing the "worked out" example. Tonight I am going to look it over more careful when I have more time. Hower, as I said to Gusbob, what I am really looking for is a General Method for raising any non-zero integer to any complex number power.

For example: (+ or minus x) where x is any integer (or, for that matter, any real number) ^ (a + bi) where a and b are any real number and i is the square root of -1

Can you help -- as I am "lost?"

7. You are not going to find an answer that you like, because it is not simple.
You need to know how to raise e to a complex number.
You also need to know that there can be more that one value, although there is a principal value.
Here are two examples.
$2^{1+2i}=\exp((1+2i)\ln(2))=2[\cos(2\ln(2))+i\sin(2\ln(2))]$
$(-3)^{-2+3i}=\exp((-2+3i)(\ln(3)+i\pi))$.

8. 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).

9. 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 --

a. Trig.

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?

10. Correction Regarding "e"

I made mistake in calling "e" an irrational number. I am aware that it is a transcentental number. My question and confustion about "e" is still intact.

11. This is not an instructional website.
You need to find a textbook that is readable and elementy.
Here is my recomadition. Basic Complex Variables
After the first 90 pages you will be able to answer your question.

12. Originally Posted by DeanSchlarbaum
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 --

a. Trig.
6
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?
1. Correct.

2. The reason is because all complex exponentiation stems from manipulating Euler's formula $e^{i\theta} = \cos{\theta} + i\sin{\theta}$. Any other base can be turned into a base of $e$, since $a^x = e^{\log{(a^x)}} = e^{x\log{a}} = (e^{x})^{\log{a}}$.

3. I agree.

4. I have just shown you that yes, there are other things you need to know.

Anyway, to answer your original question, when doing any complex function analysis, you need to know how to write a function $f(z) = f(x + iy)$ as $u + iv$, where $u$ and $v$ are real functions of $x$ and $y$.

$f(z) = (-3)^{2 + 3i}$

$= (-3)^2(-3)^{3i}$

$= 9(-3)^{3i}$

$= 9e^{\log{[(-3)^{3i}]}}$

$= 9e^{3i\log{(-3)}}$

$= 9(e^{3i})^{\log{(-3)}}$

One thing you will need to know is that $\log{z} = \ln{|z|} + i\arg{z}$.

So $\log{(-3)} = \ln{|-3|} + i\arg{(-3)}$

$= \ln{3} + \pi i$.

So $9(e^{3i})^{\log{(-3)}} = 9(e^{3i})^{\ln{3} + \pi i}$

$= 9[e^{3i(\ln{3} + \pi i)}]$

$= 9[e^{3i\ln{3} - 3\pi}]$

$= 9e^{3i\ln{3}}e^{-3\pi}$

$= 9e^{-3\pi}\left[\cos{(3\ln{3})} + i\sin{(3\ln{3})}\right]$

$= 9e^{-3\pi}\cos{(3\ln{3})} + 9i\,e^{-3\pi}\sin{(3\ln{3})}$.

Can you see now it's written as $u + iv$?

13. Basic Complex Variables

Plato --
I ordered this book from Amazon.com after reading your post. As for your remark that MHF is "not an instructional Website" -- as a matter of fact, even though I am a very new visitor/member I find it to be very instructional. However, I take it that you mean that I should "do my homework" first and then submit questions related to specific mathematical problems. Sorry if I offeneded you. Such was not my intention.

Prove It --
Thank you for at least showing me, so-to-speak, how far I have to go before I prove (or disprove) the Reimann Hypothesis. Just kidding (sort of)! Seriously, thank you for taking so much time to delineate what I have to master before I can raise integers to complex powers. The question -- the how to do it -- is very important to me.

However, I will not post again regarding this question until and if I have specific question(s) about your entry. In other words, "mum" is the word until I know (or think I know) what I (and you) are talking about!