- #1
player1_1_1
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Homework Statement
why [tex]\frac{1}{z}[/tex] don't have derivative? i know that [tex]( \log z )'=\frac{1}{z}[/tex] so why [tex]\int \frac{1}{z} \mbox{d} z[/tex] don't exist?
1/z does not have an antiderivative because it is a complex function, meaning it involves both real and imaginary numbers. The concept of antiderivatives only applies to real functions, and since 1/z is not a real function, it cannot have an antiderivative.
Yes, 1/z can be integrated using a method called Cauchy's integral formula, which involves contour integration. However, this method is more complex and requires a good understanding of complex analysis.
Yes, there are numerical methods that can approximate the integral of 1/z, such as the trapezoidal rule or Simpson's rule. These methods involve breaking up the integral into smaller parts and approximating each part with simpler functions.
There are some special cases where 1/z may have an antiderivative, such as when the function is restricted to a specific domain, or when certain assumptions are made about the function. However, in general, 1/z does not have an antiderivative.
Understanding why 1/z doesn't have an antiderivative is important in complex analysis, as it highlights the differences between real and complex functions. It also helps in understanding the limitations of integration and the various techniques used to approximate integrals of complex functions.