The argument of the complex number \(\frac{{1 + i}}{{1 - i}},\) where \(i = \sqrt { - 1}\), is
\(\frac{\pi }{2}\)
Multiplying numerator and denominator by 1+i gives (1+i)/(1−i)=i. Its principal argument is π/2.
The contour integral $$\oint_C e^{1/z} dz$$ with C as the counter-clockwise unit circle in the z-plane is equal to
Let $$ I = \int_0^1 \frac{1}{1+t} dt + \frac{\pi i}{2} \int_0^1 \frac{1}{1+e^2} dt - i \int_0^1 \frac{1}{1+it} dt, $$ where $ t $ is a real variable and $ i = \sqrt{-1} $. The value of $ I $ is ______________