Analyzing Complex Functions: Finding Missing Components

[Matterwave]

Homework Statement

The following are pieces of analytic functions of the form f(z)=u(x,y)+iv(x,y). Find the missing pieces.

a) u(x,y)=x^2-y^2+x+2

Find v, and f.

b) v(x,y)=\sqrt{\frac{1}{2}(\sqrt{x^2+y^2}-x)}

Find u, and f.

c) f(z)=tan^{-1}(z)

Find u, v.

Homework Equations

The Attempt at a Solution

Ok, I have no idea what to even start with...I don't understand why if u is given, then v is somehow constricted... (or vice versa). What's to say that v can't be any random function? I get that if f is given, then obviously u and v are given (although even the third part completely eludes me).

Someone help me start this problem please?

BTW, the professor has posted solutions online, but I don't understand them at all, so I was hoping you guys could help.

 

[╔(σ_σ)╝]

Do you know what a harmonic conjugate is ? You need to know what it is to solve your problem.

A function is analytic on a domain iff u is the harmonic conjugate of v.

 

[Matterwave]

Uh...I know what a complex conjugate is...I don't think I've ever heard of a harmonic conjugate...

 

[╔(σ_σ)╝]

Well open up your book or look it up online. If you still need help after that then post again.

 

[Matterwave]

The professor said that the books were only recommended and not mandatory...so I thought that his lecture notes would cover everything...apparently they do not? >__> Ok wikipedia here I come...

 

[Quinzio]

Go read Wikipedia on Complex Analisys.
In the first 15 rows or so there is a strong hint for you, which will bring you to another page. There you have the solution.

 

[╔(σ_σ)╝]

Well you could also use the cauchy-riemann equations to find v when given u.

 

[Quinzio]

The professor said that the books were only recommended and not mandatory...so I thought that his lecture notes would cover everything...apparently they do not? >__> Ok wikipedia here I come...

What if you read a book that states that "book are mandatory and professors are not ?"

Find a good book, or google everything, please.

 

[Matterwave]

Ok, no need to be hostile...

I didn't get a book and it's now too late to buy a book for this problem haha. I do own a book on this stuff, but it's inaccessible to me at the moment. I will try to get a book soon.

 

[Quinzio]

No intention to be hostile... :)

 

[╔(σ_σ)╝]

Ok, no need to be hostile...

I didn't get a book and it's now too late to buy a book for this problem haha. I do own a book on this stuff, but it's inaccessible to me at the moment. I will try to get a book soon.

Haha...I like you and your humor :)

If you know the cauchy riemann equations, simply equate the derievatives and integrate to solve for u or v.

 

[Matterwave]

Ok, I get how they will help me find v from u or u from v. But what about the last one? Any hints on that? I can't simply split off the real and imaginary parts of that function in any method that I know of...

 

[╔(σ_σ)╝]

For the last one artan (z) is defined in terms of the logarithm and you may be able to split the fuction into real and imaginary parts but I am not sure if it will work out.

You could always differential arctan(z) and write down u and v for the derievative and then integrate back but there might be some issues with points where the functions are not defined.

 

[Quinzio]

Homework Statement

The following are pieces of analytic functions of the form f(z)=u(x,y)+iv(x,y). Find the missing pieces.

a) u(x,y)=x^2-y^2+x+2

Find v, and f.


BTW, the professor has posted solutions online, but I don't understand them at all, so I was hoping you guys could help.

I still don't see any solution... :)

u(x,y)=x^2-y^2+x+2

u_x= 2x+1
u_y= -2y

v_y = u_x = 2x+1
v_x = -u_y= 2y

mmmmmm... feels like v(x,y) = 2xy+y

 

[Matterwave]

Yea I got part 1 and 2...(sort of cheated on 2...looked at the solutions to see the hint of transforming to polar coordinates haha)

Thanks for the help. =)