91Ó°ÊÓ

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

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

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

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

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

Describe the Riemann surface for .$\mathit{w}\mathbf{=}\sqrt{\mathbf{z}}$

To prove that the Jacobian of the transformation is $\frac{\mathbf{∂}\left(u,v\right)}{\mathbf{∂}\left(x,y\right)}\mathbf{=}{\left|f\text{'}\left(z\right)\right|}^{\mathbf{2}}$ using Cauchy- Reimann equations.

Question: Verify that the given function is harmonic, and find a functionof which it is the real part. Hint: Use Problem 2.64. For Problem 2, see Chapter 2, Section 17, Problem 19.

$\mathbf{ln}\sqrt{{\left(1+x\right)}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}$

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

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

We have discussed the fact that a conformal transformation magnifies and rotates an infinitesimal geometrical figure. We showed that $\left|dw/dz\right|$is the magnification factor. Show that the angle of$\mathit{d}\mathit{w}\mathbf{/}\mathit{d}\mathit{z}$ is the rotation angle. Hint: Consider the rotation and magnification of an arc$\mathit{d}\mathit{z}\mathbf{=}\mathit{d}\mathit{x}\mathbf{+}\mathit{i}\mathit{d}\mathit{y}$ (of length and angle arctan $\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}$ which is required to obtain the image of dz , namely dw.

Evaluate the integrals by contour integration.

$\mathit{I}\mathbf{=}{\mathbf{∫}}_{\mathbf{0}}^{\mathbf{x}}\frac{\mathbf{c}\mathbf{o}\mathbf{s}\left(\mathrm{θ}\right)\mathbf{d}\mathbf{θ}}{\mathbf{5}\mathbf{-}\mathbf{4}\mathbf{c}\mathbf{o}\mathbf{s}\left(\mathrm{θ}\right)}$

To find that the integrals by computing residue at infinity.

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