Math Challenges

Submissions for Problem #30

Problem #30

Express the given function in terms of its real and imaginary parts. Perform this decomposition:

1) In rectangular coordinates z = x + iy, and
2) In polar coordinates z = re^(it)

\[ \displaylines{f\left(z\right)=z+\frac{1}{z}\\ 1)f\left(x,y\right)=u\left(x,y\right)+iv\left(x,y\right)\\ 2)f\left(r,\theta\right)=u\left(r,\theta\right)+iv\left(r,\theta\right)} \]

← Back to problem

zapwai

Solution: \( \displaylines{u\left(x,y\right)=x+\frac{x}{x^2+y^2},\text{ }v\left(x,y\right)=y-\frac{y}{x^2+y^2}\\ \\ u\left(r,\theta\right)=\left(r+\frac{1}{r}\right)\cos\theta,\text{ }v\left(r,\theta\right)=\left(r-\frac{1}{r}\right)\sin\theta} \)