logarithm of complex number

Member
In the context of complex number, how to prove that

1. $$\displaystyle log i +log(-1+i) \neq log i(-1+i)$$
2. $$\displaystyle log i^2 =2log i$$

ThePerfectHacker

Well-known member
In the context of complex number, how to prove that

1. $$\displaystyle log i +log(-1+i) \neq log i(-1+i)$$
2. $$\displaystyle log i^2 =2log i$$
It depends how you depend a complex logarithm. The standard definition is, for $z\not = 0$,
$$\log z = \log |z| + i\arg z$$
Where $\arg z$ angle in interval $(-\pi,\pi]$.

Note, many rules for logarithms you are used to need not work for complex logarithms.

ZaidAlyafey

Well-known member
MHB Math Helper
It depends how you depend a complex logarithm. The standard definition is, for $z\not = 0$,
$$\log z = \log |z| + i\arg z$$
Where $\arg z$ angle in interval $(-\pi,\pi]$.

Note, many rules for logarithms you are used to need not work for complex logarithms.
You mean the principle logarithm ? It is actually customary to denote that with capital A for the argument hence $Arg(z) \in (-\pi , \pi ]$.

ZaidAlyafey

Well-known member
MHB Math Helper
In the context of complex number, how to prove that

1. $$\displaystyle log i +log(-1+i) \neq log i(-1+i)$$
2. $$\displaystyle log i^2 =2log i$$
Generally we have the following

$$\displaystyle \log(z_1 z_2) = \log(z_1)+\log(z_2)$$ where $\log$ defines the multiple valued function

$$\displaystyle \log(z) = \ln|z|+i arg(z)$$

The proof is not difficult especially when we prove that

$$\displaystyle arg(z_1 z_2)= arg(z_1) +arg(z_2)$$

But remember that

$$\displaystyle Arg(z_1 z_2) \neq Arg(z_1) +Arg(z_2)$$

Can you give counter examples ?

Random Variable

Well-known member
MHB Math Helper
$\text{Log} (z_{1}z_{2}) = \text{Log}(z_{2}) + \text{Log}(z_{2})$ iff $- \pi < \text{Arg}(z_{1}) + \text{Arg} (z_{2}) \le \pi$

$\text{Log}(z_{1}^{n}) = n \text{Log}(z_{1})$ iff $-\frac{\pi}{n} < \text{Arg}(z_{1}) \le \frac{\pi}{n}$

chisigma

Well-known member
The so called 'standard definition' of the logarithm of a complex variable z is, in my opinion of course, wrong and the reason of that is explained in the following example...

http://mathhelpboards.com/calculus-10/improper-integral-involving-ln-6103.html#post28032

... where the application of such a definition conducts to an erroneous computation of a definite integral wich is solvable with elementary method. I realize however that that is a 'delicate' question and it must be discussed 'with calm and reason'...

Kind regards

$\chi$ $\sigma$

ZaidAlyafey

Well-known member
MHB Math Helper
The so called 'standard definition' of the logarithm of a complex variable z is, in my opinion of course, wrong and the reason of that is explained in the following example...

http://mathhelpboards.com/calculus-10/improper-integral-involving-ln-6103.html#post28032

... where the application of such a definition conducts to an erroneous computation of a definite integral wich is solvable with elementary method. I realize however that that is a 'delicate' question and it must be discussed 'with calm and reason'...

Kind regards

$\chi$ $\sigma$
I don't understand how is that definition questionable. If we use the branch cut for $z>0$ then having the definition

$$\displaystyle Log(z) = \ln |z|+i Arg(z)$$ where $z \in (0,2\pi ]$

Then approaching the integral from above gives $Log(z) = \ln (x)$ and $Log(z) = \ln (x) +2\pi i$ when approaching it from below .