Let $$z, w$$ be complex numbers such that $$\bar{z} + i\bar{w} = 0$$ and $$\arg zw = \pi$$. Then $$\arg z$$ equals

Given the first condition:

$$ \bar{z} + i\bar{w} = 0 $$

Isolate the conjugate of z:

$$ \bar{z} = -i\bar{w} $$

Take the complex conjugate on both sides of the equation, keeping in mind that the conjugate of a conjugate returns the original complex number, and the conjugate of $$ -i $$ is $$ i $$:

$$ z = i w $$

Multiply both sides of this equation by $$ -i $$ to express w in terms of z:

$$ -iz = -i(iw) $$

$$ -iz = w $$

$$ w = -iz $$

The second condition provides the argument of the product of z and w:

$$ \arg(zw) = \pi $$

Substitute the expression for w into the argument equation:

$$ \arg(z \cdot (-iz)) = \pi $$

$$ \arg(-i \cdot z^2) = \pi $$

Use the property of arguments where the argument of a product equals the sum of the individual arguments:

$$ \arg(-i) + \arg(z^2) = \pi $$

The argument of the purely imaginary number $$ -i $$ lies on the negative imaginary axis, which means:

$$ \arg(-i) = -\frac{\pi}{2} $$

Substitute this back into the equation:

$$ -\frac{\pi}{2} + \arg(z^2) = \pi $$

Use the argument property for powers where the exponent comes out as a multiplier:

$$ -\frac{\pi}{2} + 2\arg(z) = \pi $$

Isolate the term with the argument of z by adding $$ \frac{\pi}{2} $$ to both sides:

$$ 2\arg(z) = \pi + \frac{\pi}{2} $$

$$ 2\arg(z) = \frac{3\pi}{2} $$

Divide both sides by 2 to find the final value:

$$ \arg(z) = \frac{3\pi}{4} $$

Final Answer:

The value of the argument of z is $$ \frac{3\pi}{4} $$.