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I tried to construct the derivation of the integral C with respect to Y:

$$ \frac{\partial C}{\partial Y} = ? $$

$$ C = \frac{2}{\pi} \int_0^{\infty} Re(d(\alpha) \frac{exp(-i \cdot ln(f))}{i \alpha}) d \alpha $$

with

$$d(\alpha) = exp(i \alpha (b + ln(Y)) - u) \cdot exp(v(\alpha) + z (\alpha))$$

where ##Re()## is the real part of whatever is written in the brackets, ##i## is the imaginary number and ##exp()## is the exponential function of whatever is written in the brackets. ##d(\alpha), v(\alpha), z(\alpha)## are functions depending on ##\alpha##. However, ##v(\alpha), z(\alpha)## are not dependent on ##Y##.

How do I differentiate this complex integral? As only ##d(\alpha)## depends on ##Y##, do I just need to differentiate ##d(\alpha)## with respect to ##Y## and plug the outcome of that into my integral? I think that would be what the Leibniz rule would suggest. So like:

$$ \frac{\partial C}{\partial Y} = \frac{2}{\pi} \int_0^{\infty} Re(\frac{\partial d(\alpha)}{\partial Y} \frac{exp(-i \cdot ln(f))}{i \alpha}) d \alpha$$

Would that be correct? Because then

$$\frac{\partial d(\alpha)}{\partial Y} = \frac{i \alpha}{Y} \cdot exp(i \alpha (b + ln(Y)) - u) \cdot exp(v(\alpha) + z (\alpha)) $$

Would that be the correct way to differentiate the integral or am I doing it completely wrong?

I hope I defined everything properly, if not just let me know.

Thanks a lot for your help!