Ambiguity in the evaluation of a real integral multiplied by a complex number

Question

Let's consider the following integral, where x is a real value: $int_{0}^{1}ln(i)x^{2}dx=i(frac{pi}{2}+2kpi)int_{0}^{1}x^{2}dx$. Since $ln(i)=i(frac{pi}{2}+2kpi)$, k being an integer, what value should I consider?

Fenris · Answer

In complex analysis, the logarithm is what we call a multivalued function, and you need to do what is called a branch cut in order for it to be a well defined function. The "standard" is what we call the principal branch, often denoted $text{Log}$, where the argument is chosen within the interval $(-pi, pi]$. However, other choices may be beneficial or even necessary in certain situations.
You can read more about the complex logarithm here: https://en.wikipedia.org/wiki/Complex_logarithm

Ambiguity in the evaluation of a real integral multiplied by a complex number

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