1. ## Complex Variables

I was doing homework from by Complex Variables class, and there was one question that stumped me:

Write the complex equation $z^3+5z^2=z+3i$ as two real equations.
Source: Fundamentals of Complex Analysis for Mathematics, Science and Engineering, 2nd Ed. by Saff and Snider

I don't "see" how I can rewrite this as two real equations. I looked through the section in my text, and it said absolutely nothing regarding this.

Instead of you working out the question for me, I would appreciate some tips on how to do problems like these.

2. Originally Posted by Chris L T521
I was doing homework from by Complex Variables class, and there was one question that stumped me:

Source: Fundamentals of Complex Analysis for Mathematics, Science and Engineering, 2nd Ed. by Saff and Snider

I don't "see" how I can rewrite this as two real equations. I looked through the section in my text, and it said absolutely nothing regarding this.

Instead of you working out the question for me, I would appreciate some tips on how to do problems like these.

Interesting/weird question.

i suppose this is what they are after: write $z = x + iy$, say

your two real equations come from equating the real and imaginary parts (one equation for the real part, one for the imaginary)

3. Originally Posted by Jhevon
Interesting/weird question.

i suppose this is what they are after: write $z = x + iy$, say

your two real equations come from equating the real and imaginary parts (one equation for the real part, one for the imaginary)
It doesn't factorise nicely though, so I'd say the solutions aren't nice either...

4. Originally Posted by Jhevon
Interesting/weird question.
That's what I thought...

i suppose this is what they are after: write $z = x + iy$, say

your two real equations come from equating the real and imaginary parts (one equation for the real part, one for the imaginary)
Ah...I think I thought of that...and then disregarded it. I'll try it out and ask my professor tomorrow.

Thank you

5. Originally Posted by Prove It
It doesn't factorise nicely though, so I'd say the solutions aren't nice either...
i don't think they want you to solve or factorize anything here. just expand and group the like parts (the ones with i and the ones without). no carrying anything over the equal sign or nothing. just describing what you see without the use of z. i don't see what else they could mean by the way the question is phrased.

6. Originally Posted by Jhevon
i don't think they want you to solve or factorize anything here. just expand and group the like parts (the ones with i and the ones without). no carrying anything over the equal sign or nothing. just describing what you see without the use of z. i don't see what else they could mean by the way the question is phrased.
Doing it the way you're suggesting, I end up with these equations:

$x^3-3xy^2+5x^2-5y^2=x$
$3x^2y+10xy-y^3=y+3$

Does this look right?

7. Originally Posted by Chris L T521
That's what I thought...

Ah...I think I thought of that...and then disregarded it. I'll try it out and ask my professor tomorrow.

Thank you
Substitute $z = x + iy$ (where x and y are real) and equate real and imaginary parts:

$x^3 + 5x^2 - 3xy^2 - 5y^2 = x$ .... (1)

$(3x^2 + 10x-y^2)y = y + 3$ .... (2)