Chapter 15

Put the following complex numbers into the form \(z=x+\) iy: (a) \(\\{(-1+3 i) /(2-i)\\}\); (b) \(\\{(1+2 i) /(3-4 i)\\}+\\{(2-i) /(5 i)\\}\) (c) \([\\{5\\} /\\{(1-i)(2-i)(3-i\\}]\).

Find a polar representation of \(z_{0}=1+i\) and graph this point in the complex plane.

Find all of the nth roots of 1 . That is, find the values of \(z\) which satisfy the equation $$ z^{\mathrm{n}}=1, z \neq 0 . $$

Show that \(\mathrm{f}(\mathrm{z})=\overline{\mathrm{z}}\) is nowhere differentiable using the definition of differentiability. ( \(\bar{z}\) is the complex conjugate of \(\bar{z}\).)

The function \(\mathrm{f}(\mathrm{z})=\mathrm{z}^{2}=\mathrm{x}^{2}-\mathrm{y}^{2}+\mathrm{i} 2 \mathrm{xy}=\mathrm{u}(\mathrm{x}, \mathrm{y})+\mathrm{iv}(\mathrm{x}, \mathrm{y})\) can be shown to be differentiable for all \(\mathrm{z}\) with $$ \\{\mathrm{d} / \mathrm{d} z\\}\left(z^{2}\right)=2 z $$ Prove that this function satisfies the Cauchy-Riemann Theorem for all \(\mathrm{z}\).

Prove that the function \(\mathrm{f}(\mathrm{z})=\\{1 / \mathrm{z}\\}\) is differentiable for all \(z\) except \(z=0\) by using the Cauchy-Riemann equations. Also find \(\mathrm{f}^{\prime}(\mathrm{z})\).

Let \(\mathrm{f}(z)=0\) \(z=0\) and \(=\left[\left(x^{3}-y^{3}\right) /\left(x^{2}-y^{2}\right)\right]+\left[\left\\{i\left(x^{3}+y^{3}\right)\right\\} /\left\\{x^{2}+y^{2}\right\\}\right], \quad z \neq 0\) Show that the Cauchy-Riemann equations are satisfied at \(z=0\) but that \(\mathrm{f}^{\prime}(0)\) does not exist.

Find the harmonic conjugates of the function $$ u(x, y)=y^{3}-3 x^{2} y $$

Find the real and imaginary parts of \(\sin z\) for complex \(z\). That is, write \(\sin z\) in the form \(\operatorname{Re}(\sin z)+i \operatorname{Im}(\sin z)\).

Determine whether the following equation is true for all complex z: $$ \log \mathrm{e}^{z}=z $$

Determine the values of \(\log (1-i)\) and specify the principal value.

Define branches for the multiple-valued complex function of \(\mathrm{z}, \mathrm{z}^{1 / \mathrm{n}}\). Also determine the branch point and the principal branch of this function, (n is a positive integer.)

Justify the definition \(\mathrm{e}^{\mathrm{ix}}=\cos \mathrm{x}+\mathrm{i} \sin \mathrm{x}\) by the formal manipulation of the real power series expansions of these functions.

Expand the function \(\mathrm{f}(\mathrm{z})=\left\\{1 / \mathrm{z}^{2}\right\\}\) in a complex Taylor series about the point \(\mathrm{z}_{0}=2 .\) What is the largest circle centered at \(\mathrm{z}_{0}\) in the interior of which this expansion is valid?

Obtain series expansions of the function $$ \mathrm{f}(\mathrm{z})=[\\{-1\\} /\\{(\mathrm{z}-1)(\mathrm{z}-2)\\}] $$ about the point \(\mathrm{z}_{0}=0\) in the regions $$ |z|<1,1<|z|<2,|a|>2 $$

Find the principal part of the function \(\mathrm{f}(\mathrm{z})=\left\\{\left(\mathrm{e}^{z} \cos \mathrm{z}\right) /\left(\mathrm{z}^{3}\right)\right\\}\) at its singular point and determine the type of singular point it is.

Find the principal part of the function $$ \mathrm{f}(\mathrm{z})=\left[\\{z\\} /\left\\{(z+1)^{2}\left(\mathrm{z}^{3}+2\right)\right\\}\right] $$ at \(z_{0}=-1\). What type of singular point is \(z_{0}\) ?

Evaluate the following definite integrals where \(t\) is a real number: a) \(2 \int_{1}\left(\mathrm{t}+\mathrm{it}^{2}\right) \mathrm{dt}\); b) \({ }^{1} \int_{0} \mathrm{e}^{(\mathrm{a}+\mathrm{bi}) \mathrm{t}} \mathrm{dt}\).

Evaluate \(^{1} \int_{-1} z^{1 / 2} d z\) where \(z^{1 / 2}=\sqrt{\mid z} \mid e^{i \theta / 2}\) where \(0<\theta<2 \pi\) along any contour lying in the upper half plane and also along any contour lying in the lower half plane. Use indefinite integrals to do this.

Evaluate the integral \(\int_{C}\left[\\{z \mathrm{z} z\\} /\left\\{\left(9-z^{2}\right)(z+i)\right\\}\right]\) where \(\mathrm{C}\) is the circle \(|\mathrm{z}|=2\) described in the positive sense.

Use the residue theorem to evaluate $$ \int_{C}[\\{5 z-2\\} /\\{z(z-1)\\}] d z $$ where \(\mathrm{C}\) is the circle \(|\mathrm{z}|=2\) described counterclockwise.

Use the residue theorem to evaluate $$ \int_{C}\left[\left\\{\left(1+z^{5}\right) \sinh z\right\\} /\left\\{z^{6}\right\\}\right] d z $$ where \(\mathrm{C}\) is the unit circle \(|z|=1\) described in the positive sense (i.e., counterclockwise).

Evaluate \(\int_{C}\left[\left\\{z e^{z}\right\\} /\left\\{z^{2}-1\right\\}\right] d z\) where \(C\) is the circle \(|z|=2\) taken in the counterclockwise direction.

Find the partial fraction expansion of $$ f(z)=\left\\{\left(z^{2}+1\right) /\left(z^{3}+4 z^{3}+3 z\right)\right\\} $$ using the theory of residues.

Evaluate \(\mathrm{I}=2 \pi \int_{0}\\{1 /(\cos \theta+2)\\} \mathrm{d} \theta\)

Evaluate \(\mathrm{I}=\pi \int_{-\pi} \mathrm{e}^{2 \cos \theta} \mathrm{d} \theta\)

Evaluate \(\mathrm{I}_{0}={ }^{\infty} \int_{0}\left\\{\mathrm{~d} \mathrm{x} /\left(\mathrm{x}^{2}+1\right)\right\\}\)

Evaluate \(\mathrm{I}_{0}={ }^{\infty} \int_{0}\left[\left\\{\mathrm{x}^{2} \mathrm{~d} \mathrm{x}\right\\} /\left\\{\left(\mathrm{x}^{2}+9\right)\left(\mathrm{x}^{2}+4\right)^{2}\right\\}\right]\).

Evaluate \(I_{1}=^{\infty} \int_{-\infty}\left\\{(x \cos x) /\left(x^{2}+1\right)\right\\} \mathrm{d} x\) and \(I_{2}={ }^{\infty} \int_{-\infty}\left\\{(\mathrm{x} \sin \mathrm{x}) /\left(\mathrm{x}^{2}+1\right)\right\\} \mathrm{d} \mathrm{x}\).