Full solution: Integrdte by the first method or state why it does not apply and then use

Find and sketch the path and its orientation given by: \(z(t)=(1+3 i) t(1 \leqq t \leqq 4)\) Text Transcription: z(t) = (1 + 3i) t(1 leqq t leqq 4)

Find and sketch the path and its orientation given by: \(z(t)=5-2 it (-3 \leqq t \leqq 3)\) Text Transcription: z(t) = 5 - 2it (-3 leqq t leqq 3)

Find and sketch the path and its orientation given by: \(z(t)=4+i+3 e^{i t}(0 \leqq t \leqq 2 \pi)\) Text Transcription: z(t) = 4 + i + 3e^{it} (0 leqq t leqq 2 pi)

Find and sketch the path and its orientation given by: \(z(t)=1+i+e^{-\pi i t}(0 \leqq t \leqq 2)\) Text Transcription: z(t) = 1 + i + e^{-pi it} (0 leqq t leqq 2)

Find and sketch the path and its orientation given by: \(z(t)=e^{i t}(0 \leqq t \leqq \pi)\) Text Transcription: z(t) = e^{it} (0 leqq t leqq pi)

Find and sketch the path and its orientation given by: \(z(t)=3+4 i+5 e^{i t}(\pi \leqq t \leqq 2 \pi)\) Text Transcription: z(t) = 3 + 4i + 5e^{it} (pi leqq t leqq 2 pi)

Find and sketch the path and its orientation given by: \(z(t)=6 \cos 2 t+5 i \sin 2 t(0 \leqq t \leqq \pi)\) Text Transcription: z(t) = 6 cos 2t + 5i sin 2 t(0 leqq t leqq pi)

Find and sketch the path and its orientation given by: \(z(t)=1+2 t+8 i t^{2}(-1 \leqq t \leqq 1)\) Text Transcription: z(t) = 1 + 2t + 8i t^2 (-1 leqq t leqq 1)

Find and sketch the path and its orientation given by: \(z(t)=t+\frac{1}{2} i t^{3}(-1 \leqq t \leqq 2)\) Text Transcription: z(t) = t +1 / 2 i t^3(-1 leqq t leqq 2)

Sketch and represent parametrically: Segment from 1 + i to 4 - 2i

Sketch and represent parametrically: Unit circle (clockwise)

Sketch and represent parametrically: Segment from a + ib to c + id

Sketch and represent parametrically: Hyperbola xy = 1 from 1 + i to \(4+\frac{1}{4} i\) Text Transcription: 4 + 1 / 4 i

Sketch and represent parametrically: Semi-ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1, y \geqq 0\) Text Transcription: x^2/a^2 + y^2/b^{2} = 1, y geqq 0

Sketch and represent parametrically: 15. Parabola \(y=4-4 x^{2}(-1 \leqq x \leqq 1)\) Text Transcription: y = 4 - 4x^2(-1 leqq x leqq 1)

Sketch and represent parametrically: |z - 2 + 3i| = 4 (counterclockwise)

Sketch and represent parametrically: |z + a + ib| = r (clockwise)

Sketch and represent parametrically: Ellipse \(4(x-1)^{2}+9(y+2)^{2}=36\) Text Transcription: 4(x - 1)^2 + 9(y + 2)^2 = 36

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C} \operatorname{Re} z d z, C\) the shortest path from 0 to 1 + i Text Transcription: int_C Re z dz, C

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C} \operatorname{Re} z d z, C\) the parabola \(y=x^{2}\) from 0 to 1 + i Text Transcription: int_C Re z dz, C y = x^2

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C} e^{2 z} d z, C\) the shortest path from \(\pi i\) to \(2 \pi i\) Text Transcription: int_C e^{2z} dz, C pi i 2pi i

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C} \sin z d z, C\) any path from 0 to 2i Text Transcription: int_C sin z dz, C

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C} \cos ^{2} z d z\) from \(-\pi i\) along \(|z|=\pi\) to \(\pi i\) in the right half-plane Text Transcription: int_C cos^2 z dz -pi i |z| = pi pi i

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C}\left(z+z^{-1}\right) d z, C\) the unit circle (counterclockwise) Text Transcription: int_C(z + z^-1) dz, C

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C} \cosh 4 z d z, C\) any path from \(-\pi i / 8\) to \(\pi i / 8\) Text Transcription: int_C cosh 4z dz, C -pi i/8 pi i/8

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C} \bar{z} d z, C\) from -1 + i along the parabola \(y=x^{2}\) to 1 + i Text Transcription: int_C bar{z} dz, C y = x^2

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C} \sec ^{2} z d z, C\) any path from \(\pi / 4\) to \(\pi i / 4\) Text Transcription: int_C sec^2 z dz, C pi/4 pi i/4

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C} \operatorname{Im}_{z_{1}}^{2} d z\) countercolockwise around the triangle with vertices z = 0, 1, i Text Transcription: int_C Im_z_1^{2} dz

Integrate by the first method or state why it does not apply and then use the second method. (Show the details of your work.) \(\int_{C} z e^{z^{2} / 2} d z, C\) from i along the axes to 1 Text Transcription: int_C z e^z^2/2 dz, C

Verify (5) for \(f(z)=z^{2}\), where C is the segment from -1- i to 1 + i. Text Transcription: f(z) = z^2

Verify (6) for f(z) = 1/z and \(C_{1} and \(C_{2}\) the upper and lower halfs of the unit circle Text Transcription: C_1 C_2

(ML-inequality) Find an upper bound of the absolute value of the integral in Prob. 19.

(Linearity) Illustrate (4) with an example of your own. Prove (4).

Integration. (a) Comparison. Write a short report comparing the essential points of the two integration methods. (b) Comparison. Evaluate \(\int_{C} f(z) d z\) by Theorem 1 and check the result by Theorem 2, where: (i) \(f(z)=z^{4}\) and C is the semicircle |z| = 2 from -2i to 2i in the right half-plane, (ii) \(f(z)=\mathrm{e}^{2 z}\) and C is the shortest path from 0 to 1 + 2i. (c) Continuous deformation of path. Experiment with a family of paths with common endpoints, say, \(z(t)=t+i a \sin t, 0 \leqq t \leqq \pi\), with real parameter a. Integrate nonanalytic functions \(\left(\operatorname{Re} z, \operatorname{Re}\left(z^{2}\right)\right.\), etc.) and explore how the result depends on a. Then take analytic functions of your choice. (Show the details of your work.) Compare and comment. (d) Continuous deformation of path. Choose another family, for example, semi-ellipses \(z(t)=a \cos t+i \sin t,-\pi / 2 \leqq t \leqq \pi / 2\), and experiment as in (c). Text Transcription: int_C f(z) dz f(z) = z^4 f(z) = e^{2z} z(t) = t + ia sin t, 0 leqq t leqq pi (Re z, Re(z^2) etc) z(t) = a cos t + i sin t, - pi/2 leqq t leqq pi/2

CAS PROJECT. Integration. Write programs for the two integration methods. Apply them to problems of your choice. Could you make them into a joint program that also decides which of the two methods to use in a given case?