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SomeBody
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- It looks like the complex function 1/(z-p) does not comply with the Cauchy-Goursat theorem and the defination of the complex logarithm for certain values of z.
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid.
I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem:
Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of ##D## then:
$$F(z)=\int_{z_{0}}^{z} f(w)dw$$
is analytic over ##D## and ##F'(z)=f(z)##
Now the question:
If I pick ##f(w)=\frac{1}{z-p}## and a region ##D## with ##z_{0}## in it but without point ##p## enclosed (think some u-shape region stretched around p on one side) then ##f## is analytic over ##D## and I should be able to apply the theorem above. So:
$$F(z)=\int_{z_{0}}^{z} \frac{1}{z-p}dw=ln\left(\frac{z-p}{z_{0}-p} \right)$$
Now let ##z_{1}=-z_{0}+2p## be in ##D## which gives the following result:
$$F(z_{1})=ln\left(\frac{-z_{0}+2p-p}{z_{0}-p} \right)=ln\left(-1 \right)$$
However in the first chapter of my syllabus the complex logarithm is defined as:
$$Ln(z)=ln(r)+i\varphi$$ for every ##z=re^{i\varphi}## with ##\varphi\neq \pi+2k\pi, k\in \mathbb{Z}## and undefined for complex numbers with negative real part and zero imaginary part (like for the negative real numbers).
So if ##F(z_{1})## is undefined, how can ##F## be analytical over ##D## like the theorem states?
I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem:
Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of ##D## then:
$$F(z)=\int_{z_{0}}^{z} f(w)dw$$
is analytic over ##D## and ##F'(z)=f(z)##
Now the question:
If I pick ##f(w)=\frac{1}{z-p}## and a region ##D## with ##z_{0}## in it but without point ##p## enclosed (think some u-shape region stretched around p on one side) then ##f## is analytic over ##D## and I should be able to apply the theorem above. So:
$$F(z)=\int_{z_{0}}^{z} \frac{1}{z-p}dw=ln\left(\frac{z-p}{z_{0}-p} \right)$$
Now let ##z_{1}=-z_{0}+2p## be in ##D## which gives the following result:
$$F(z_{1})=ln\left(\frac{-z_{0}+2p-p}{z_{0}-p} \right)=ln\left(-1 \right)$$
However in the first chapter of my syllabus the complex logarithm is defined as:
$$Ln(z)=ln(r)+i\varphi$$ for every ##z=re^{i\varphi}## with ##\varphi\neq \pi+2k\pi, k\in \mathbb{Z}## and undefined for complex numbers with negative real part and zero imaginary part (like for the negative real numbers).
So if ##F(z_{1})## is undefined, how can ##F## be analytical over ##D## like the theorem states?
