PLAGUE
- 35
- 2
- TL;DR Summary
- Why we replace y with 0 and why this is equivalent to replacing x with (z + z')/2 and y with (z - z')/2i?
I am given, $$u = e^{-x} (x sin y - y cos y)$$ and asked to find v such that, $$f(z) = u + iv$$. My book solves these problems and the answer is, $$v = e^{-x} (ysiny + x cos y) + c$$. I understand how it is done, using Cauchy-Riemann equations.
Then, the book asks to find f(z). When doing that, it says, $$f(z) = f(x+iy) = u(x, y) + iv(x, y)$$
Putting y = 0, $$f(x) = u(x, 0) + iv(x, 0)$$
Replacing x by z, $$f(z) = u(z, 0) + iv(z, 0)$$
The answer: $$f(z) = ize^{-z}$$
Why is replacing y with 0 is valid here?
The book solves this method in another method. It replaces x with $$\frac{z + \overline z}{2}$$ and y with $$\frac{z - \overline z}{2i}$$. Then, it simplifies and the answer is still the same, $$i z e^{-z}$$.
Why is this method equivalent to the previous one?
N.B. I know that you get the real part of z by $$\frac{z + \overline z}{2}$$ and the imaginary part by $$\frac{z - \overline z}{2i}.$$
I am giving the full answer here. They are taken from
by Murray SPIEGEL (Author)
Then, the book asks to find f(z). When doing that, it says, $$f(z) = f(x+iy) = u(x, y) + iv(x, y)$$
Putting y = 0, $$f(x) = u(x, 0) + iv(x, 0)$$
Replacing x by z, $$f(z) = u(z, 0) + iv(z, 0)$$
The answer: $$f(z) = ize^{-z}$$
Why is replacing y with 0 is valid here?
The book solves this method in another method. It replaces x with $$\frac{z + \overline z}{2}$$ and y with $$\frac{z - \overline z}{2i}$$. Then, it simplifies and the answer is still the same, $$i z e^{-z}$$.
Why is this method equivalent to the previous one?
N.B. I know that you get the real part of z by $$\frac{z + \overline z}{2}$$ and the imaginary part by $$\frac{z - \overline z}{2i}.$$
I am giving the full answer here. They are taken from
by Murray SPIEGEL (Author)
Last edited: