Complex analysis: domains, contours and proof obligations
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 210 minutes, as an unvalidated planning estimate. This collection contains 11 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Choose methods before opening guidance. Write the domain, assumptions and intermediate reasoning; a decimal answer or theorem name alone is insufficient.
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A germ at a continues along every path in a domain D. For a fixed-endpoint homotopy H, one row has three compact segment margins 1/5,1/8,1/6 and seam radii 1/10,1/20. Give a strict perturbation tolerance, construct finite compatible strips and a common refinement, and prove endpoint-germ invariance without assuming a single function on the image of H. If D is simply connected, prove local holomorphy of the endpoint assignment. As changed cases: start the cube root of z−2 at 3 with value1 and continue it around 2 once; then start the square root of z²−1 at2 with value√3 and compare loops 2exp(2πit) and 1+exp(2πit). Identify what each computation does and does not establish.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
At z=−i, compare logarithm values using arguments −π/2 and 3π/2, and their associated square roots. Continue a logarithm along z(t)=2e^(it), 0≤t≤4π, and compute ∫dz/z. Explain why the endpoint cannot determine the continued value.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For f(z)=conj(z)², determine every point where the complex derivative exists. Verify the Cauchy–Riemann equations there and explain why this does not make f holomorphic on any neighborhood of zero.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For 1/z on 1<|z|<4, compare the integrals over the positively oriented circles |z|=2 and |z|=3 and over their oriented difference. Explain precisely why Cauchy’s theorem does not make each integral zero, and describe the key estimate in Goursat’s triangle proof.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compute ∮ e^z/(z−2)³ dz around |z|=3 counterclockwise. Then prove that an entire function satisfying |f(z)|≤C(1+|z|³) is a polynomial of degree at most three.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Find the domain of convergence and sum of Σ((z−1)/3)^n. If a holomorphic function on a connected domain containing 1 vanishes at 1+1/n for all sufficiently large n, what follows? Contrast zeros at positive integers for an entire function, and explain why a nonconstant holomorphic function cannot have an interior local maximum of its modulus.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Expand 1/[(z−1)(z−3)] about zero in both annuli 1<|z|<3 and |z|>3. Explain the different z⁻¹ coefficients and the corresponding counterclockwise circle integrals, noting that the original function is holomorphic at zero. Compare the singularities of (1−cos z)/z² and exp(1/z²) at zero.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Evaluate ∫ over the real line of 1/(x²+9)² dx using an upper semicircle, including a bound showing that the arc contribution vanishes.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For P(z)=z⁵+3z+1, count zeros inside |z|=1/2 and |z|=2 and hence in the intervening annulus. Compute ∮F′/F dz on |z|=2 for F(z)=P(z)/(z−1)².
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show that T(z)=(z−2)/(z+2) maps the right half-plane bijectively to the unit disk. Find its inverse and derivative at 2, and explain the boundary point corresponding to infinity.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Find a harmonic conjugate of u(x,y)=x²−y²+2y and recover u on the unit disk from its boundary Fourier data. State a uniqueness argument. Contrast this with log|z| on an annulus and explain the role of topology in continuing its conjugate.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
A germ at a continues along every path in a domain D. For a fixed-endpoint homotopy H, one row has three compact segment margins 1/5,1/8,1/6 and seam radii 1/10,1/20. Give a strict perturbation tolerance, construct finite compatible strips and a common refinement, and prove endpoint-germ invariance without assuming a single function on the image of H. If D is simply connected, prove local holomorphy of the endpoint assignment. As changed cases: start the cube root of z−2 at 3 with value1 and continue it around 2 once; then start the square root of z²−1 at2 with value√3 and compare loops 2exp(2πit) and 1+exp(2πit). Identify what each computation does and does not establish.
At z=−i, compare logarithm values using arguments −π/2 and 3π/2, and their associated square roots. Continue a logarithm along z(t)=2e^(it), 0≤t≤4π, and compute ∫dz/z. Explain why the endpoint cannot determine the continued value.
For f(z)=conj(z)², determine every point where the complex derivative exists. Verify the Cauchy–Riemann equations there and explain why this does not make f holomorphic on any neighborhood of zero.
For 1/z on 1<|z|<4, compare the integrals over the positively oriented circles |z|=2 and |z|=3 and over their oriented difference. Explain precisely why Cauchy’s theorem does not make each integral zero, and describe the key estimate in Goursat’s triangle proof.
Compute ∮ e^z/(z−2)³ dz around |z|=3 counterclockwise. Then prove that an entire function satisfying |f(z)|≤C(1+|z|³) is a polynomial of degree at most three.
Find the domain of convergence and sum of Σ((z−1)/3)^n. If a holomorphic function on a connected domain containing 1 vanishes at 1+1/n for all sufficiently large n, what follows? Contrast zeros at positive integers for an entire function, and explain why a nonconstant holomorphic function cannot have an interior local maximum of its modulus.
Expand 1/[(z−1)(z−3)] about zero in both annuli 1<|z|<3 and |z|>3. Explain the different z⁻¹ coefficients and the corresponding counterclockwise circle integrals, noting that the original function is holomorphic at zero. Compare the singularities of (1−cos z)/z² and exp(1/z²) at zero.
Show that T(z)=(z−2)/(z+2) maps the right half-plane bijectively to the unit disk. Find its inverse and derivative at 2, and explain the boundary point corresponding to infinity.
Find a harmonic conjugate of u(x,y)=x²−y²+2y and recover u on the unit disk from its boundary Fourier data. State a uniqueness argument. Contrast this with log|z| on an annulus and explain the role of topology in continuing its conjugate.
