Separate Sine Function with Complex Numbers

[KleZMeR]

Homework Statement

I'd like to separate this function to U(x) + i*V(y) form. It's a homework problem that is asking if it is an analytic function. Searching thru trig substitutions, but looking ahead I don't see much luck...
Any suggestions or help is greatly appreciated.

Homework Equations

The Attempt at a Solution

 

[Ray Vickson]

G

Homework Statement

I'd like to separate this function to U(x) + i*V(y) form. It's a homework problem that is asking if it is an analytic function. Searching thru trig substitutions, but looking ahead I don't see much luck...
Any suggestions or help is greatly appreciated.

Homework Equations

The Attempt at a Solution

Please read the "pinned" post Guidelines for Students and Helpers" by Vela, which explains the standards expected in this Forum, and in particular, why you should not post thumbnails. They are OK for things like diagrams from books, etc, but not for work. Type out the statements of problems, and your solutions.

Such thumbnails are not readable on some media, and make it difficult for people (who freely volunteer their time) to offer help.

 

[KleZMeR]

Thanks Ray. I totally understand, and I apologize for the obscurity. To be honest and a bit anecdotal, I got a heavy flu the second week of grad school (last week) and missed class and lost 3 days of homework time, so I'm just trying to catch up. The equations I'm dealing with are long so typing them out is a bit time-consuming in my crunch, but I will totally adhere to the forum rules, I apologize again, and really do appreciate all the help I receive here. Here's my question below. Let me know if it needs a new thread.

Homework Statement

Consider complex function F(z) = sin(αZ^2) , where a = α+iβ is a constant. Is this function analytic, entire, discuss the differentiability.

Homework Equations

Euler Identity for sin(x) was tried but I don't see how to separate the Real and Imaginary. Should I use DeMoivre's theorem? Or a combination of some Identity and DeMoivre?

The Attempt at a Solution

Even if this function has a conjugate harmonic, I am still having trouble getting it into a form:
F(z) = U(x) + i*V(x).

Been searching for a useful Trig Identity to separate the Imaginary and Real parts of the 'angle' but to no Avail. I hope this is a sufficient attempt.

 

[KleZMeR]

Ok, I found a useful identity, going to try it!

 

[KleZMeR]

Further analysis

OK, so, I figured it out, but the answer has raised a question regarding whether it is analytic.

I found:

F (z) = U(x,y) + iV(x,y) = sin(x)cosh(y) + i*cos(x)sinh(y)

Where in the previous problem #1, I was given U(x,y) = sin(x)cosh(y) , and told to find the harmonic conjugate V, which in turn generated:

F (z) = U(x,y) + iV(x,y) = sin(x)cosh(y) + i*cos(x)*sinh(y) (same function!)

Although I assumed that this function was analytic because it has the conjugate, the previous problem simplified to:

F(z) = sin(z*) , which tells me that dF/dz* ≠ 0 , and I think this violates the Cauchy-Riemann relation.

Given this I would say it is not analytic, and I'm 99% sure of this, but since this is my first work of this sort any encouragement helps.

The instructor actually left a hint for us at the end of the problem saying that (to answer question #2 and #3, I may want to use the expression resulting from #1, which was F(z)=sin(z*).

 

[KleZMeR]

And to solidify my results, Ux≠Vy and Uy≠Vx , they are off by a unit negative factor I found.

 

[vela]

Are you talking about $F(z) = \sin(a z^2)$ still? That function is analytic.

 

[KleZMeR]

Yea, I'm correcting my mistake, ugh. I was wrong on my integration, I had a function show up rather than a constant.

I ended up with V(x,y) = -cos(x)sinh(y) , but it's actually V(x,y)=cos(x)sinh(y) + constant.

This small algebraic error left me with sin(Z*) , which is wrong. Thanks vela, I appreciate the confirmation.

 

[Ray Vickson]

Thanks Ray. I totally understand, and I apologize for the obscurity. To be honest and a bit anecdotal, I got a heavy flu the second week of grad school (last week) and missed class and lost 3 days of homework time, so I'm just trying to catch up. The equations I'm dealing with are long so typing them out is a bit time-consuming in my crunch, but I will totally adhere to the forum rules, I apologize again, and really do appreciate all the help I receive here. Here's my question below. Let me know if it needs a new thread.

Homework Statement

Consider complex function F(z) = sin(αZ^2) , where a = α+iβ is a constant. Is this function analytic, entire, discuss the differentiability.

Homework Equations

Euler Identity for sin(x) was tried but I don't see how to separate the Real and Imaginary. Should I use DeMoivre's theorem? Or a combination of some Identity and DeMoivre?

The Attempt at a Solution

Even if this function has a conjugate harmonic, I am still having trouble getting it into a form:
F(z) = U(x) + i*V(x).

Been searching for a useful Trig Identity to separate the Imaginary and Real parts of the 'angle' but to no Avail. I hope this is a sufficient attempt.

Below, I will simplify the typing by using $\alpha = a + ib$ instead of your $a = \alpha + i \beta$.

If all you want to know is whether
$$f(z) = \sin \left(\alpha z^2 \right)$$
is analytic (or, at least, holomorphic) you do not need to find the real and imaginary parts, although doing so cannnot hurt. Is $w = \alpha z^2$ analytic in $z$? Are $e^{iw}, \: e^{-iw}$ analytic in $w$? Recall that
$$\sin(w) = \frac{1}{2i} \left( e^{iw} - e^{-iw} \right).$$

Alternatively: we have
$$w = \alpha z^2 = [a(x^2-y^2)-2bxy]+i[b(x^2-y^2)+2axy] \equiv u + iv$$
Use
$$\sin(w) = \frac{1}{2i} \left( e^{iu - v} - e^{-iu + v} \right)\\<br /> \text{and}\\<br /> e^{iu-v} = e^{-v} \left(\cos(u) + i \sin(u) \right),\:<br /> e^{-iu+v} = e^v \left( \cos(u) - i \sin(u) \right).$$
By grinding it through you can eventually obtain the real and imaginary parts $U(x,y), V(x,y)$ that you want.

 

[KleZMeR]

Thanks Ray, these points definitely help my understanding of how to approach this problem. I just finished another one for:

sin(pi*(x-iy)^2) which I believe is analogous to sin(Z*)^2

And as you said, if I say: W=(Z*)^2 and I ask if this is analytic, it is not because dW/dZ* ≠ 0.

Therefore sin(Z*)^2 is not analytic either.

 

[Ray Vickson]

Thanks Ray, these points definitely help my understanding of how to approach this problem. I just finished another one for:

sin(pi*(x-iy)^2) which I believe is analogous to sin(Z*)^2

And as you said, if I say: W=(Z*)^2 and I ask if this is analytic, it is not because dW/dZ* ≠ 0.

Therefore sin(Z*)^2 is not analytic either.

Right: the complex conjugate $\bar{z}$ is not an analytic function of $z$. The Cauchy-Riemann equations fail when $u(x,y) = x, \: v(x,y) = -y$.