91Ó°ÊÓ

In equation (7.18), let u (x) be an even function and $\mathit{Ï\dots }\left(x\right)$be an odd function.

1. If $\mathit{f}\left(x\right)\mathbf{=}\mathit{u}\left(x\right)\mathbf{+}\mathit{i}\mathit{Ï\dots }\left(x\right)$, show that these conditions are equivalent to the equation${\mathit{f}}^{\mathbf{*}}\left(x\right)\mathbf{=}\mathit{f}\left(-x\right)$ .
2. Show that

$\mathit{Ï€}\mathit{u}\left(a\right)\mathbf{=}\mathit{P}\mathit{V}\underset{\mathbf{0}}{\overset{\mathbf{∞}}{\mathbf{∫}}}\frac{\mathbf{2}\mathbf{x}\mathbf{Ï\dots }\left(x\right)}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{a}}^{\mathbf{2}}}\mathbf{ }\mathit{d}\mathit{x}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathit{Ï€}\mathit{Ï\dots }\left(a\right)\mathbf{=}\mathbf{-}\mathit{P}\mathit{V}\underset{\mathbf{0}}{\overset{\mathbf{∞}}{\mathbf{∫}}}\frac{\mathbf{2}\mathbf{a}\mathbf{u}\left(x\right)}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{a}}^{\mathbf{2}}}\mathbf{ }\mathit{d}\mathit{x}$

These are Kramers-Kroning relations. Hint: To find u(a), write the integral for u(a) in (7.18) as an integral from $\mathbf{-}\mathbf{∞}$to 0 plus an integral from 0 to $\mathbf{∞}$. Then in the to integral $\mathbf{-}\mathbf{∞}$to 0, replace x by -x to get an integral from 0 to$\mathbf{∞}$ , and userole="math" localid="1664350095623" $\mathit{Ï\dots }\left(-x\right)\mathbf{=}\mathbf{-}\mathit{Ï\dots }\left(x\right)$ . Add the two to integrals and simplify. Similarly findrole="math" localid="1664350005594" $\mathit{Ï\dots }\mathbf{\left(}\mathbf{a}\mathbf{\right)}$ .

w =√z. Hint: This is equivalent to w² = z; find x and y in terms of u and v and then solve the pair of equations for u and v in terms of x and y. Note that this is really the same problem as Problem 1 with the z and w planes interchanged.

Use the Cauchy-Riemann conditions to find out whether the functions in Problems 1.1 to 1.21 are analytic.

${\mathit{e}}^{\mathbf{z}}$

To find: uand v as a function of x and y & plot the graph and show curve u = constant constant should be orthogonal to the curves v = constant . w = sin z

Find the real and imaginary parts $\mathit{u}\left(x,y\right)$ and $\mathit{v}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$of the following functions.

$\mathit{c}\mathit{o}\mathit{s}\mathit{h}\mathit{z}$

Find out whether infinity is a regular point, an essential singularity, or a pole (and if a pole, of what order) for each of the following functions. Find the residue of each function at infinity,$\frac{\mathbf{4}{\mathbf{z}}^{\mathbf{3}}\mathbf{+}\mathbf{2}\mathbf{z}\mathbf{+}\mathbf{3}}{{\mathbf{z}}^{\mathbf{2}}}$

To find u and vas a function of xand y & plot the graph and show curve u = constant should be orthogonal to the curves v =constant.

w = cosh z

Find the real and imaginary parts $\mathit{u}\left(x,y\right)$ and $\mathit{v}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$of the following functions.

$\mathit{s}\mathit{i}\mathit{n}\mathit{z}$

Describe the Riemann surface for w = z³

Find the real and imaginary parts $\mathit{u}\left(x,y\right)$and $\mathbf{v}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$of the following functions.

$\frac{\mathbf{1}}{\mathbf{z}}$