On the contour segment from A=(0+i) to B=(1+i) (where y=1, x:0to1, dZ=dx), with f(Z)=y-x-3ix^2, the contribution to the contour integral is:

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CSS Pure Mathematics QUESTION #4099

Question 1

On the contour segment from \(A=(0+i)\) to \(B=(1+i)\) (where \(y=1\), \(x:0\to1\), \(dZ=dx\)), with \(f(Z)=y-x-3ix^2\), the contribution to the contour integral is:

\(\dfrac{1}{2}-i\)✔️
\(-\dfrac{1}{2}+i\)
\(1-i\)
\(\dfrac{1}{2}-\dfrac{i}{2}\)

Correct Answer Explanation

On \(AB\): \(y=1\), \(Z=x+i\), \(dZ=dx\), \(f=1-x-3ix^2\). Integral: \(\int_0^1(1-x-3ix^2)dx = \left[x-\dfrac{x^2}{2}-ix^3\right]_0^1 = 1-\dfrac{1}{2}-i = \dfrac{1}{2}-i\).

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