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I Complex integration

  1. May 22, 2017 #1
    Hello! I started learning about complex analysis and I am a bit confused about integration. I understand that if we take different paths for the same function, the value on the integral is different, depending on the path. But if we use the antiderivative: ##\int_{\gamma}f=F(\gamma(b))-F(\gamma(a))##, where ##\gamma## is the path and a and b are the endpoints. So based on this formula, the value of the integral doesn't depends on the path but just on the endpoints. I am not sure I understand the meaning of this and when can we use this formula, as it gives just a value for all the path, so what do the other paths means? Thank you!
     
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  3. May 22, 2017 #2

    BvU

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    The antiderivative expression holds if ##F## is complex differentiable -- which is a severe constraint. See holomorphic functions
     
  4. May 22, 2017 #3
    But if I take ##f(z)=z^2##, this has the antiderivative ##F=(z)=\frac{z^3}{3}##, which is entire. So if I take the path from ##0## to ##1+i##, using this formula I obtain ##\int_{\gamma}f=F(i+1)-F(0)=\frac{2i-2}{3}##. However, if I use the formula ##\int_{\gamma}f=\int_a^b f(\gamma(t))\gamma'(t)dt## and I parametrize ##\gamma## using a straight line or a parabola between the same 2 points (##0## and ##1+i##), I obtain 2 different results. So in the first case only the end points matter, in the second case, the path matters, too. So how am I suppose to use the antiderivative formula?
     
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