Determine the location and kind of the singularities of the following functions in the

Determine the location and kind of the singularities of the following functions in the finite plane and at infinity. In the case of poles also state the order. \(\tan ^{2} \pi z\) Text Transcription: tan^2 pi z

Determine the location and kind of the singularities of the following functions in the finite plane and at infinity. In the case of poles also state the order. \(z+\frac{2}{z}-\frac{3}{z^{2}}\) Text Transcription: z + 2 / z - 3 / z^2

Determine the location and kind of the singularities of the following functions in the finite plane and at infinity. In the case of poles also state the order. \(\cot z^{2}\) Text Transcription: cot z^2

Determine the location and kind of the singularities of the following functions in the finite plane and at infinity. In the case of poles also state the order. \(z^{3} e^{1 /(z-1)}\) Text Transcription: z^{3} e^{1/(z - 1)}

Determine the location and kind of the singularities of the following functions in the finite plane and at infinity. In the case of poles also state the order. cos z - sin z

Determine the location and kind of the singularities of the following functions in the finite plane and at infinity. In the case of poles also state the order. 1/(cos z - sin z)

Determine the location and kind of the singularities of the following functions in the finite plane and at infinity. In the case of poles also state the order. \(\frac{\sin 3 z}{\left(z^{4}-1\right)^{4}}\) Text Transcription: sin 3 z / (z^4 - 1)^4

Determine the location and kind of the singularities of the following functions in the finite plane and at infinity. In the case of poles also state the order. \(\frac{4}{z-1}+\frac{2}{(z-1)^{2}}-\frac{8}{(z-1)^{3}}\) Text Transcription: 4 / z - 1 + 2 / (z - 1)^2 - 8 / (z - 1)^3

Determine the location and kind of the singularities of the following functions in the finite plane and at infinity. In the case of poles also state the order. \(\cosh \left[1 /\left(z^{2}+1\right)\right]\) Text Transcription: cosh [1/(z^2 + 1)]

Determine the location and kind of the singularities of the following functions in the finite plane and at infinity. In the case of poles also state the order. \(e^{1 /(z-1)} /\left(e^{z}-1\right)\) Text Transcription: e^1/(z - 1)/(e^{z} - 1)

(Essential singularity) Discuss \(e^{1 / z^{2}}\) in a similar way as \(e^{1 / z}\) is discussed in Example 3. Text Transcription: e^{1/z^2} e^1/z

(Poles) Verify Theorem 1 for \(f(z)=z^{-3}-z^{-1}\). Prove Theorem 1. Text Transcription: f(z) = z^ -3 -z^{-1}

Determine the location and order of the zeros. \((z+16 i)^{4}\) Text Transcription: (z + 16i)^4

Determine the location and order of the zeros. \(\left(z^{4}-16\right)^{4}\) Text Transcription: (z^4 - 16)^4

Determine the location and order of the zeros. \(z^{-3} \sin ^{3} \pi z\) Text Transcription: z^{-3} sin^3 pi z

Determine the location and order of the zeros. \(\cosh ^{2} z\) Text Transcription: cosh^{2} z

Determine the location and order of the zeros. \(\left(3 z^{2}+1\right) e^{-z}\) Text Transcription: (3z^2 + 1) e^{-z}

Determine the location and order of the zeros. \(\left(z^{2}-1\right)^{2}\left(e^{z^{2}}-1\right)\) Text Transcription: (z^2 - 1)^{2}(e^{z^{2} - 1)

Determine the location and order of the zeros. \(\left(z^{2}+4\right)\left(e^{z}-1\right)^{2}\) Text Transcription: (z^2 + 4) (e^{z} - 1)^2

Determine the location and order of the zeros. \((sin z-1)^{3}\) Text Transcription: (sin z - 1)^3

Determine the location and order of the zeros. \((1-\cos z)^{2}\) Text Transcription: (1 - cos z)^2

Determine the location and order of the zeros. \(e^{z}-e^{2 z}\) Text Transcription: e^z - e^2z

If f(z) is analytic and has a zero of order n at \(z={z}_{0}\). show that \(f^{2}(z)\) has a zero of order 2n. Text Transcription: z = z_0 f^2 (z)

Zeros. (a) Derivative. Show that if f(z) has a zero of order n > 1 at \(z={z}_{0}\). then f ‘ (z) has a zero of order n - 1 at \(z_{0}\). (b) Poles and zeros. Prove Theorem 4. (c) Isolated k - points. Show that the points at which a nonconstant analytic function f(z) has a given value k are isolated. (d) Identical functions. If \(f_{1}(z)\) are analytic in a domain D and equal at a sequence of points \(z_{n}\) in D that converges in D, show that \(f_{1}(z) \equiv f_{2}(z)\) in D. Text Transcription: z = z_0 z_0 f_1(z) z_n f_1(z) equiv f_{2}(z)

Assuming that we let the image of the x-axis be meridians \(0^{\circ}\) and \(180^{\circ}\), describe and sketch (or graph) the images of the following regions on the Riemann sphere: (a) |z| > 100, (b) the lower half-plane, (c) \(\frac{1}{2} \leqq|z| \leqq 2\). Text Transcription: 0^circ 180^circ 1 / 2 leqq |z| leqq 2