The same is true for integrals: if you take
\(
\int \frac1x \,dx
\)
you can get different functions wich differ not only by a constant. The convention here is to count the solution as \( \ln |x|+C \) but this is only one of possible solutions. Another for example is \( \ln |x|+ \operatorname{sgn}(x)+C \)
You can also claim that there are two solutions ln x and ln (-x) which suitable only for one half of the domain of definition. In fact neither of them is the solution for the function on the whole real axis.
By analogy with logarithm it is reasonable to count that
\( \sum_z \frac{1}{z + 1} = \psi(|z+\frac12|+\frac12)+C \)
and
\( \sum_z \frac{1}{z} = \psi(|z-\frac12|+\frac12)+C \).
This is a graph of how it looks like:
![[Image: h_1286684374_10064fb295.png]](http://static.itmages.ru/i/10/1010/h_1286684374_10064fb295.png)
Just use absolute values and you'll get a compact form for the solution. For integrals the conventional form is determined by the Cauchy principal value.
\(
\int \frac1x \,dx
\)
you can get different functions wich differ not only by a constant. The convention here is to count the solution as \( \ln |x|+C \) but this is only one of possible solutions. Another for example is \( \ln |x|+ \operatorname{sgn}(x)+C \)
You can also claim that there are two solutions ln x and ln (-x) which suitable only for one half of the domain of definition. In fact neither of them is the solution for the function on the whole real axis.
By analogy with logarithm it is reasonable to count that
\( \sum_z \frac{1}{z + 1} = \psi(|z+\frac12|+\frac12)+C \)
and
\( \sum_z \frac{1}{z} = \psi(|z-\frac12|+\frac12)+C \).
This is a graph of how it looks like:
Just use absolute values and you'll get a compact form for the solution. For integrals the conventional form is determined by the Cauchy principal value.