Thread: Integrating complex logs

Hi,

I'm having trouble getting my head around integrating a function such as:

f(z)=log(z+2),

where z is a complex number.

Any ideas on where to start?

Thanks

Tony

Originally Posted by Tony2710

Hi,

I'm having trouble getting my head around integrating a function such as:

f(z)=log(z+2),

where z is a complex number.

Any ideas on where to start?

Thanks

Tony

How about $\displaystyle (z+2)\log(z+2) - (z+2) + \alpha$ where $\displaystyle \alpha\in \mathbb{C}$ ?
(Treat this as a regular calculus problem).

Originally Posted by Tony2710

Hi,

I'm having trouble getting my head around integrating a function such as:

f(z)=log(z+2),

where z is a complex number.

Any ideas on where to start?

Thanks

Tony

Yea, plot it. But first plot Log(z) then draw some contours over the real and imaginary surfaces and interpret:

$\displaystyle \mathop\int\limits_{C} Log(z)dz$

and:

$\displaystyle \oint Log(z)dz$

I personally think until you understand the geometry of a multifunction (it's real and imaginary Riemann surfaces), contour integrations over these surfaces will always be difficult to understand. Take time to draw them and understand them, then the integrals become a breeze.

I want to add something to what I said. The function $\displaystyle \log (z+5)$ has a primitive $\displaystyle g$ on $\displaystyle \mathbb{C} - (-\infty,-5]$. On this "slit plane" (or cut-plane) $\displaystyle g'(z) = \log (z+5)$. However, on $\displaystyle (-\infty,-5]$ the primitive is not even continous (because the logarithm jumps its value as it comes around). Therefore, if you are trying to compute the line integral from $\displaystyle A$ to $\displaystyle B$ and the curve that you are integration does not cross over $\displaystyle (-\infty,-5]$ then the integral is simply $\displaystyle g(B) - g(A)$. However, if the curve does cross over then you need to be careful and cannot use this fundamental theorem for line integrals.