Complex, holomorphic and integration

[Potage11]

1.Represent the following rational functions as sums of elementary fractions and find the primitive functions (indefinite integrals);
(a) f(z)= z-2/z^2 +1


2.Now, I am pretty sure that I have to split up this equation into real and imaginary parts in order to show that it is holomorphic. Once I have show that, I can use that fact to find a g'(z)=f(z)


3. The Attempt at a Solution :

So I tried f(z)=f(x+iy)= x+iy-2/(x+iy)^2 +1
u= x-2/(x+iy)^2 +1 and v= y/(x+iy)^2 +1

but by Cauchy-Reimann I am not getting du/dx=dv/dy, even though I know I should.
du/dx= -x^2 -y^2 +4x+1+4iy and
dv/dy= x^2 + y^2 +1

so I got frustrated, and attempted to do the integral anyways, first by substitution, then by parts;
This is my best attempt so far:

Int[z/z^2+1] + Int[-2/z^2+1]
Int[z/z^2+1]=1/2ln|z^2+1|
Int[-2/z^2+1]= Int[-2/(z+i)(z-i)]
I don't feel like listing everything, but by partial fractions:
Int[-2/z^2+1]= Int[i/z-i] + Int[-i/z-i]= iln(z-i) - iln(z+i)

Final answer: 1/2ln|z^2+1|+ iln(z-i) - iln(z+i) +c

I would like to know if my answer makes even remote sense, and if you guys can figure out why my cauchy-reimann equation isn't working, that would be awesome too.

 

[Dick]

Your integral looks reasonable. The reason Cauchy-Riemann isn't working is that if you are writing f(x+iy)=u(x,y)+v(x,y)*i, u and v should be real functions. You haven't gotten rid of all of the i's in them.

 

[SiddharthM]

i'm not sure what constitutes a elementary function (i always thought rational functions were elementary!).

But the reason your not getting the cauchy reimann equations satisfied, well look at how you split up u(x,y) and v(x,y)

u= x-2/(x+iy)^2 +1 and v= y/(x+iy)^2 +1

Notice something wrong? Remember that the C-R equations work when u and v are of the form f(x,y)=u(x,y)+iv(x,y), where u and v are REAL functions of two variables. Your u and v above have complex numbers in them!

Where is the function:

g(z) = 1/2ln|z^2+1|+ iln(z-i) - iln(z+i) +c

differentiable? We know that wherever it IS differentiable, it's derivative is your original function - but the problem is asking for the subset of the plane under which this is true.

 

[Potage11]

Thanks Dick, that's such a morale boost!

So I'm trying to remove the i's:
I multiplied out x+iy-2/x^2 - y^2 + 2ixy +1
I'm pretty sure I can't do u=x-2/x^2 - y^2 +1 and v=y/2xy
I am honestly out of ideas :(

 

[Dick]

If you have a fraction like c/(a+bi), to make the denominator real, multiply top and bottom by (a-bi). That will put all of the i's in the numerator where you can collect them.

 

[Potage11]

Just for fun, Maple gave me 2/z+ln(z)+z for that integral.
How it got there, I have no idea, but it looks much cleaner :)

 

[Potage11]

omg, I can't believe I forgot the conjugate... thanks again

 

[Potage11]

Ok, so that took forever, but it pretty much worked, thanks for the help guys!

 

[Dick]

Just for fun, Maple gave me 2/z+ln(z)+z for that integral.
How it got there, I have no idea, but it looks much cleaner :)

If you differentiate that, do you get what you started with? I don't think so. I think you typed something wrong into Maple.