Thread: Finding the solutions with complex power

1. Finding the solutions with complex power

$\displaystyle \displaystyle\ 3e^{4xi}-12e^{2xi+x}+45e^{2xi+2x}+3=0$
I would like to find the solution of this equation, but i do not even know which is the 'leading' term, as in the one with the highest power of t. How would i compare $\displaystyle \displaystyle\ t^{4i}$ to $\displaystyle \displaystyle \ t^{2i+2}$to atleast find which is the leading term? How could i go about solving this?

2. It definitely belongs in pre-university, but that's not to say it's not tricky.

First, you need to remember what i stands for; the square root of -1. This is an imaginary number, which means we can't deal with it without going into advanced stuff. You can, however, drastically simplify this problem. Remember that:

$\displaystyle 3t^{4i} = (3t^4)^i$

Think you can go from here?

3. 3 isn't risen to the power of 4i, only t is. So with that being the case, i am guessing you ment $\displaystyle \displaystyle 3(t^4)^i$

With that in mind, are you suggesting i substitute $\displaystyle \displaystyle t^{2i}=u$and then somehow treat it as a quadratic?The problem i find with that is what do $\displaystyle \displaystyle t, t^2 and t^{-1}$ get substituted with?

4. Originally Posted by BariMutation
It definitely belongs in pre-university, but that's not to say it's not tricky. $\displaystyle 3t^{4i} = (3t^4)^i$
I totally disagree with that quote.
Any problem dealing with complex exponentiation is definitely a university level: perhaps number theory or analysis.

Complex exponents do not behave as expected.
For example, if each of $\displaystyle u,~v,~\&~w$ is a complex number then $\displaystyle [u^v]^w\not=u^{vw}$.
Only in the case that $\displaystyle n\in \mathbb{Z}$ can we say that $\displaystyle [u^v]^n\not=u^{nv}$

Thus as applied to this problem it can be said that $\displaystyle t^{4i}=[t^i]^4$.
Here is a further compilation: $\displaystyle t^i=\exp[i\log(t)]$ where $\displaystyle \log(t)=\ln(|t|)+i\;\text{arg}(t)$.

It is not clear if t is a complex variable or a real variable.

5. Originally Posted by Plato
I totally disagree with that quote.
Any problem dealing with complex exponentiation is definitely a university level: perhaps number theory or analysis.

Complex exponents do not behave as expected.
For example, if each of $\displaystyle u,~v,~\&~w$ is a complex number then $\displaystyle [u^v]^w\not=u^{vw}$.
Only in the case that $\displaystyle n\in \mathbb{Z}$ can we say that $\displaystyle [u^v]^n\not=u^{nv}$

Thus as applied to this problem it can be said that $\displaystyle t^{4i}=[t^i]^4$.
Here is a further compilation: $\displaystyle t^i=\exp[i\log(t)]$ where $\displaystyle \log(t)=\ln(|t|)+i\;\text{arg}(t)$.

It is not clear if t is a complex variable or a real variable.
I was wondering the same... I tried to make a substitution y = t^i, but I ended up with y and t together and this held me back from posting here. I'll be waiting for the solution to this question, once the OP comes to clarify the question, it seems really interesting!

6. Originally Posted by Plato
I totally disagree with that quote.
Here is a further compilation: $\displaystyle t^i=\exp[i\log(t)]$ where $\displaystyle \log(t)=\ln(|t|)+i\;\text{arg}(t)$.

It is not clear if t is a complex variable or a real variable.
I assume exp stands for exponential? I must also admit i have never used the arg(t) function.
Perhaps i should clarify this:

This is not a problem i found in a textbook, in fact it is an equation deriving from a function's intersection with the Ox axis.

Background: I solved a differential equation and found the form of a function y(x). As i was curious, i attempted to draw the graph of that function, which was in exponential form and with complex powers. In order to draw the graph, i wanted to see where the function intersected the Oy and Ox axis. In an attempt to find where it intersected the Ox axis, i made it y(x)=0. I worked to try to simplify the equation in order to find the solutions and eventually reached what has been presented as the above equation.

I should note that, in order to simplify and make things clearer for me, i substituted $\displaystyle \displaystyle t=e^x$. That is where the t comes from.

With that being said, i expected it not be the average problem. The reason i put it in Pre-University was because it wasn't calculus or differentiation and had nothing to do with those, it was an equation and i did not find a more suitable place to post it. I was an am still curious to how this can be solved for the sheer pleasure of it

7. Originally Posted by kamykazee
I should note that, in order to simplify and make things clearer for me, i substituted $\displaystyle \displaystyle t=e^x$. That is where the t comes from.
Now are you saying that $\displaystyle t^{4i}$ is really $\displaystyle e^{4xi}~?$
If so rewrite the whole problem as it should be.

8. Here is the rewritten equation:

$\displaystyle \displaystyle\ 3e^{4xi}-12e^{x(2i+1)}+45e^{2x(i+1)}+3=0$

Could anyone aid me in how it could be solved?

9. Originally Posted by kamykazee
Here is the rewritten equation:
$\displaystyle \displaystyle\ 3e^{4xi}-12e^{x(2i+1)}+45e^{2x(i+1)}+3=0$
In the future please post the question not what you think we need to know.
Two of us have wasted time on this.

In general if $\displaystyle x~\&~y$ are real numbers then $\displaystyle e^{x+yi}=e^x\left(\cos(y)+i~\sin(y)\right)$
Example: $\displaystyle e^{x(2i+1)}=e^x\left(\cos(2x)+i~\sin(2x)\right)$

10. Thank you for your answers. I'm sorry i was not explicit enough in the beginning and for posing this question in an inappropriate place. $\displaystyle \displaystyle |x|$