# Thread: Cauchy's integral formula help

1. ## Cauchy's integral formula help

Q: Using Cauchy’s integral formula, compute the integral of g(z) over the circle of radius 3 centred at the origin, the contour integral being taken counterclockwise.

g(z) =z^3 + z^2 − 5/z − 2

i cant wrap my head around this formula or how to even begin solving it. all other examples had two 'poles' rather tha just (z-2)

any help will be much appreciated.

2. Can you set up the integral?
What does Cauchy's integral formula tell you?

By the way, it's not clear if you meant $z^3+z^2-5/z -2$ or $z^3+z^2-\frac{5}{z -2}$.

3. sorry im not sure what you mean by set up.
to be clear its (z^3+z^2-5)/(z-2)
I know i have to use the denominator to find the singularities of g(z) but i do not know how to plug in the equation into the formula.

4. Originally Posted by EoinCahill
sorry im not sure what you mean by set up.
to be clear its (z^3+z^2-5)/(z-2)
I know i have to use the denominator to find the singularities of g(z) but i do not know how to plug in the equation into the formula.
That's a long shot from what you had written. You do know that $a+b \times c \neq (a+b) \times c$, right? Parentheses are not a luxury.

It's clear that the only singularity is at $z=2$.
Now can you write $g(z)dz$ in the form $\frac{f(z)}{z-2}dz$, where $f$ is holomorphic inside the contour? What does Cauchy's formula tell you about the integral of such a form along the contour?

5. It's clear that the only singularity is at z=2.
yes i thought that myself.
Now can you write in the form , where is holomorphic inside the contour?
so f(z) = z^3+z^2-5??

that the integral is equal to 2.pi.i??

6. Originally Posted by EoinCahill
yes i thought that myself.

so f(z) = z^3+z^2-5??

that the integral is equal to 2.pi.i??
Yes. No.

You need to review Cauchy's Integral formula (see post #4 here for example (in your question n = 1): http://www.mathhelpforum.com/math-he...uchys-thm.html)

7. ok this is what i have so far.

f(z)= ( z^3+z^2-5 dz
------) z-2

is this right???

8. Originally Posted by EoinCahill
ok this is what i have so far.

f(z)= ( z^3+z^2-5 dz
------) z-2

is this right???
I said in my earlier reply that yes your f(z) was correct and no your answer was not correct. Where has the above come from - it is nothing more than a re-statement of the question! Did you read the post I refered you to? You need to compare the given integral to Cauchy's Integral Formula and:

1. Identify f(z).

2. Identify $\alpha$.

Then you need to:

3. Evaluate $f(\alpha)$.