Expressing Complex Functions in Real and Imaginary Components
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2. In each case, write the function $f(z)$ in the form $f(z) = u(x, y) + iv(x, y)$:
(a) $f(z) = z^3 + z + 1$;
(b) $f(z) = \frac{\bar{z}^2}{z} \quad (z \neq 0)$.
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Step by Step Written Solution
In this problem, we want to express the complex function f of z in its standard form, u of x comma y plus i times v of x comma y, where u is the real part and v is the imaginary part. Let's start with part a.
Complex Function Decomposition
For part a, we have f of z equals z cubed plus z plus one. To find the real and imaginary components, we substitute z equals x plus i y.
Part (a)
Plugging this in, the expression becomes x plus i y cubed plus x plus i y plus one.
Let's expand the cube using the binomial theorem. It gives us x cubed, plus three x squared times i y, plus three x times i y squared, plus i y cubed.
Recall that i squared is negative one and i cubed is negative i. Simplifying the terms, we get x cubed plus three i x squared y, minus three x y squared, minus i y cubed.
Now let's put it all back into the full expression for f of z.
We can now identify the real part u and the imaginary part v.
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