Functions of a Complex Variable

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.

$\mathbf{Re}\mathit{z}$

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 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.

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

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.

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

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.

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

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.

$\mathit{c}\mathit{o}\mathit{s}\overline{\mathbf{z}}$

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

$\sqrt{\mathbf{z}}$

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

$\overline{\mathbf{z}}$

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

$\left|z\right|$

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

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

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

$\frac{2z+3}{z+2}$

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

$\frac{2z+3}{z+2}$

Using the definition (2.1) of , show that the following familiar formulas hold. Hint : Use the same methods as for functions of a real variable.

26..

Using the definition (2.1) of , show that the following familiar formulas hold. Hint : Use the same methods as for functions of a real variable.

27..

Using the definition (2.1) of , show that the following familiar formulas hold. Hint : Use the same methods as for functions of a real variable.

28.. (See hint below.)

Problem 28 is the chin rule for the derivative of a function of a function.

Using the definition (2.1) of show that the following familiar formulas hold.

Differentiate Cauchy’s formula (3.9) or (3.10) to get 

$\mathbf{f}\mathbf{\text{'}}\left(\mathbf{z}\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{\pi i}}{\oint }_{\mathbf{C}}\frac{\mathbf{f}\left(\mathbf{w}\right)\mathbf{dw}}{{\left(\mathbf{w}\mathbf{-}\mathbf{z}\right)}^{\mathbf{2}}}$ or $\mathbf{f}\mathbf{\text{'}}\left(\mathbf{a}\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{\pi i}}{\oint }_{\mathbf{C}}\frac{\mathbf{f}\left(\mathbf{z}\right)\mathbf{dz}}{{\left(\mathbf{z}\mathbf{-}\mathbf{\alpha }\right)}^{\mathbf{2}}}$

By differentiating n times, obtain

${\mathbf{f}}^{\left(\mathbf{n}\right)}\left(\mathbf{z}\right)\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{2}\mathbf{\pi i}}{\oint }_{\mathbf{C}}\frac{\mathbf{f}\left(\mathbf{w}\right)\mathbf{dw}}{{\left(\mathbf{w}\mathbf{-}\mathbf{z}\right)}^{\mathbf{n}\mathbf{+}\mathbf{1}}}$ or ${\mathbf{f}}^{\left(\mathbf{n}\right)}\left(\mathbf{\alpha }\right)\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{2}\mathbf{\pi i}}{\oint }_{\mathbf{C}}\frac{\mathbf{f}\left(\mathbf{z}\right)\mathbf{dz}}{{\left(\mathbf{z}\mathbf{-}\mathbf{a}\right)}^{\mathbf{n}\mathbf{+}\mathbf{1}}}$

Show that equation (4.4) can be written as (4.5). Then expand each of the fractions in the parenthesis in (4.5) in powers of z and in powers of $\frac{\mathbf{1}}{\mathbf{z}}$ [see equation (4.7) ] and combine the series to obtain (4.6), (4.8), and (4.2). For each of the following functions find the first few terms of each of the Laurent series about the origin, that is, one series for each annular ring between singular points. Find the residue of each function at the origin. (Warning: To find the residue, you must use the Laurent series which converges near the origin.) Hints: See Problem 2. Use partial fractions as in equations (4.5) and (4.7). Expand a term $\frac{\mathbf{1}}{\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{\alpha }\mathbf{)}}$ in powers of z to get a series convergent for $\left|\mathbf{z}\right|\mathbf{<}\mathbf{\alpha }$, and in powers of $\frac{\mathbf{1}}{\mathbf{z}}$  to get a series convergent for $\left|\mathbf{z}\right|>\mathbf{\alpha }$.

For each of the following functions find the first few terms of each of the Laurent series about the origin, that is, one series for each annular ring between singular points. Find the residue of each function at the origin. (Warning: To find the residue, you must use the Laurent series, which converges near the origin.) Hints: See Problem 2. Use partial fractions as in equations (4.5) and (4.7). Expand a term $\frac{\mathbf{1}}{\left(\mathbf{z}\mathbf{-}\mathbf{\alpha }\right)}$  in powers of z to get a series convergent for $\left|\mathbf{z}\right|\mathbf{<}\mathbf{\alpha }$, and in powers of  $\frac{\mathbf{1}}{\mathbf{z}}$ to get a series convergent for $\left|\mathbf{z}\right|\mathbf{>}\mathbf{\alpha }$.

 $\mathbf{f}\left(\mathbf{z}\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{z}\left(\mathbf{z}\mathbf{-}\mathbf{1}\right)\left(\mathbf{z}\mathbf{-}\mathbf{2}\right)}$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

 $\frac{{\mathbf{e}}^{\mathbf{2}\mathbf{z}}}{\mathbf{1}\mathbf{+}{\mathbf{e}}^{\mathbf{z}}}\mathbf{\text{ at }}\mathit{z}\mathbf{=}\mathit{i}\mathit{\pi }$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

 $\frac{{\mathbf{e}}^{\mathbf{i}\mathbf{z}}}{\mathbf{9}{\mathbf{z}}^{\mathbf{2}}\mathbf{+}\mathbf{4}}\mathbf{\text{ at }}\mathit{z}\mathbf{=}\frac{\mathbf{2}\mathbf{i}}{\mathbf{3}}$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

 $\frac{\mathbf{1}\mathbf{-}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{2}\mathbf{z}}{{\mathbf{z}}^{\mathbf{3}}}\mathbf{\text{ at }}\mathit{z}\mathbf{=}\mathbf{0}$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

 $\frac{{\mathbf{e}}^{\mathbf{2}\mathbf{z}}\mathbf{-}\mathbf{1}}{{\mathbf{z}}^{\mathbf{2}}}\mathbf{\text{ at }}\mathit{z}\mathbf{=}\mathbf{0}$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

 $\frac{{\mathbf{e}}^{\mathbf{2}\mathbf{\pi iz}}}{\mathbf{1}\mathbf{-}{\mathbf{z}}^{\mathbf{3}}}$at  $\mathbf{z}\mathbf{=}{\mathbf{e}}^{\frac{\mathbf{2}\mathbf{\pi i}}{\mathbf{3}}}$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

$\frac{\mathbf{cosz}}{\mathbf{1}\mathbf{-}\mathbf{2}\mathbf{sinz}}$ at $\mathbf{z}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{6}}$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

$\frac{\mathbf{z}\mathbf{+}\mathbf{2}}{\left({\mathbf{z}}^{\mathbf{2}}\mathbf{+}\mathbf{9}\right)\left({\mathbf{z}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\right)}$ at $\mathbf{z}\mathbf{=}\mathbf{3}\mathbf{i}$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

 $\frac{{\mathbf{e}}^{\mathbf{2}\mathbf{z}}}{\mathbf{4}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{h}\mathbf{z}\mathbf{-}\mathbf{5}}$at $\mathit{z}\mathbf{=}\mathit{l}\mathit{n}\mathbf{2}$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

$\frac{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{z}\mathbf{-}\mathbf{1}}{{\mathbf{z}}^{\mathbf{7}}}$ at z = 0

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

$\frac{{\mathbf{e}}^{\mathbf{3}\mathbf{z}}\mathbf{-}\mathbf{3}\mathbf{z}\mathbf{-}\mathbf{1}}{{\mathbf{z}}^{\mathbf{4}}}$ at z = 0

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

$\frac{{\mathbf{e}}^{\mathbf{iz}}}{{\mathbf{(}{\mathbf{z}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathbf{)}}^{\mathbf{2}}}\mathbf{ }\mathbf{at}\mathbf{ }\mathbf{z}\mathbf{=}\mathbf{2}\mathbf{i}$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

$\frac{\mathbf{1}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{z}}{{\mathbf{(}\mathbf{\pi }\mathbf{-}\mathbf{z}\mathbf{)}}^{\mathbf{2}}}$ at $\mathit{z}\mathbf{ }\mathbf{=}\mathit{\pi }$

Find the inverse Laplace transform of the following functions using (7.16) $\frac{{\mathbf{p}}^{\mathbf{3}}}{{\mathbf{p}}^{\mathbf{4}}\mathbf{+}\mathbf{4}}$.

Find the inverse Laplace transform of the following functions by using (7.16) $\frac{\mathbf{1}}{{\mathbf{p}}^{\mathbf{4}}\mathbf{-}\mathbf{1}}$.

Find the inverse Laplace transform of the following functions by using (7.16). $\frac{\mathbf{p}\mathbf{+}\mathbf{1}}{\mathbf{p}\left(p^2+1\right)}$

Find the inverse Laplace transform of the following functions by using (7.16).

$\frac{\mathbf{p}}{\left(p+1\right)\left(p^2+4\right)}$

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

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

$\mathit{\pi }\mathit{u}\left(a\right)\mathbf{=}\mathit{P}\mathit{V}\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}\frac{\mathbf{2}\mathbf{x}\mathbf{\upsilon }\left(x\right)}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{a}}^{\mathbf{2}}}\mathbf{ }\mathit{d}\mathit{x}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathit{\pi }\mathit{\upsilon }\left(a\right)\mathbf{=}\mathbf{-}\mathit{P}\mathit{V}\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}\frac{\mathbf{2}\mathbf{a}\mathbf{u}\left(x\right)}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{a}}^{\mathbf{2}}}\mathbf{ }\mathit{d}\mathit{x}$

These are Kramers-Kroning relations. Hint: To find u(a), write the integral for u(a) in (7.18) as an integral  from $\mathbf{-}\mathbf{\infty }$to 0 plus an integral from 0 to $\mathbf{\infty }$. Then in the to integral $\mathbf{-}\mathbf{\infty }$ to 0, replace x by -x to get an integral from 0 to $\mathbf{\infty }$ , and use  $\mathit{\upsilon }\left(-x\right)\mathbf{=}\mathbf{-}\mathit{\upsilon }\left(x\right)$ . Add the two to integrals and simplify. Similarly find  $\mathit{\upsilon }\mathbf{\left(}\mathbf{a}\mathbf{\right)}$ .

Let f(z)  be expanded in the Laurent series that is valid for all  z outside some circle, that is, $\left|z\right|\mathbf{>}\mathit{M}$(see Section 4). This series is called the Laurent series "about infinity." Show that the result of integrating the Laurent series term by term around a very large circle (of  radius > M) in the positive direction, is  $\mathbf{2}\mathit{\pi }\mathit{i}{\mathit{b}}_{\mathbf{1}}$ (just as in the original proof of the residue theorem in Section 5). Remember that the integral "around $\mathbf{\infty }$   " is taken in the negative direction, and is equal to  $\mathbf{2}\mathit{\pi }\mathit{i}$ : (residue at  $\mathbf{\infty }$ ). Conclude that $\mathit{R}\mathbf{(}\mathbf{\infty }\mathbf{)}\mathbf{=}\mathbf{-}{\mathit{b}}_{\mathbf{1}}$ . Caution: In using this method of computing $\mathit{R}\mathbf{(}\mathbf{\infty }\mathbf{)}$  be sure you have the Laurent series that converges for all sufficiently large  z.

(a) Show that if f(z)  tends to a finite limit as z  tends to infinity, then the residue of f(z)  at infinity is.

(b) Also show that if  f(z) tends to zero as z  tends to infinity, then the residue of f(z)  at infinity is  $\mathbf{-}\mathbf{lim}\mathbf{ }{\mathbf{z}}^{\mathbf{2}}\mathbf{f}\mathbf{\text{'}}\mathbf{\left(}\mathbf{z}\mathbf{\right)}$.

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,  $\frac{\mathbf{z}}{{\mathbf{z}}^{\mathbf{2}}\mathbf{+}\mathbf{1}}$.

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, $\frac{\mathbf{2}\mathbf{z}\mathbf{+}\mathbf{3}}{{\left(z+2\right)}^{\mathbf{2}}}$.

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, $\mathit{s}\mathit{i}\mathit{n}\mathbf{ }\frac{\mathbf{1}}{\mathbf{z}}$ .

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, $\frac{\mathbf{4}{\mathbf{z}}^{\mathbf{3}}\mathbf{+}\mathbf{2}\mathbf{z}\mathbf{+}\mathbf{3}}{{\mathbf{z}}^{\mathbf{2}}}$

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, $\frac{\mathbf{1}\mathbf{+}\mathbf{z}}{\mathbf{1}\mathbf{-}\mathbf{z}}$

Evaluate the following integrals by computing residues at infinity. Check your answers by computing residues at all the finite poles. (It is understood that $\mathbf{\oint }$  means in the positive direction.)

$\mathbf{\oint }\frac{\mathbf{1}\mathbf{-}{\mathbf{z}}^{\mathbf{2}}}{\mathbf{1}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}}\frac{\mathbf{d}\mathbf{z}}{\mathbf{z}}$  around  $\left|z\right|\mathbf{=}\mathbf{2}$

To find that the integrals by computing residue at infinity.

 ${\mathbf{\oint }}_{\mathbf{c}}\frac{{\mathbf{z}}^{\mathbf{2}}}{\left(2z+1\right)\left(z^2+9\right)}$ around $\left|z\right|\mathbf{=}\mathbf{5}$ .

To prove that the sum of the residues at finite points plus the residence at infinity is zero.

For each of the following functions w = f(z) = u +iv, find u and v as functions of x and y. Sketch the graph in (x,y) plane of the images of u = const. and v = const. for several values of and several values of as was done for in Figure 9.3. The curves u = const. should be orthogonal to the curves v = const.

w = e^z

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.

To find:  u and 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

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

w = cosh z