Equation of a vertical line in the complex plane

zeebo17
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Homework Statement



I need to solve \int_L \bar{z}-1 where L is the line from 1 to 1+2i.


Homework Equations





The Attempt at a Solution



I know that I need to set z equal to the equation of the line and then integrate, but in this case I'm not sure how to express the equation of the vertical line in the complex plane.

In the case of the non-vertical line I would write z=x+i(mx+b) where m is the slope of the line and b is the y-intercept, however in this case the y is undefined.
 
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zeebo17 said:

Homework Statement



I need to solve \int_L \bar{z}-1 where L is the line from 1 to 1+2i.


Homework Equations





The Attempt at a Solution



I know that I need to set z equal to the equation of the line and then integrate, but in this case I'm not sure how to express the equation of the vertical line in the complex plane.

In the case of the non-vertical line I would write z=x+i(mx+b) where m is the slope of the line and b is the y-intercept, however in this case the y is undefined.

The y-values aren't undefined; they're arbitrary. Wouldn't your equation be z = 1 + iy, 0 <= y <= 2i?
 
Oh! Great, thanks!
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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