MATH-20 · Advanced · ENGLISH · PLANNED
Build the picture, explain the rule, work an example, then justify it.
Study time is an initial activity estimate, excluding extra reading, revision and extended laboratory or project work. Videos and question bodies have not been produced by adding this outline.
Complete these courses or demonstrate equivalent knowledge. These are recommended prerequisites, not enforced enrolment locks.
YouTube playlist: ARK Mathematics | Complex Analysis and Fourier Methods
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Sign in to save this learning pathPractical outcome: Compare real and complex local transformations.
Explain complex plane using a visual and one worked real-world example. Context: Compare real and complex local transformations. End with a prediction or check-for-understanding.
Explain complex exponential using a visual and one worked real-world example. Context: Compare real and complex local transformations. End with a prediction or check-for-understanding.
Explain holomorphic functions using a visual and one worked real-world example. Context: Compare real and complex local transformations. End with a prediction or check-for-understanding.
Explain cauchy-riemann equations using a visual and one worked real-world example. Context: Compare real and complex local transformations. End with a prediction or check-for-understanding.
Use the first concepts from this section. Compare real and complex local transformations. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain complex logarithm branches using a visual and one worked real-world example. Context: Compare real and complex local transformations. End with a prediction or check-for-understanding.
Explain contour integrals using a visual and one worked real-world example. Context: Compare real and complex local transformations. End with a prediction or check-for-understanding.
Explain cauchy's theorem using a visual and one worked real-world example. Context: Compare real and complex local transformations. End with a prediction or check-for-understanding.
Explain cauchy integral formula using a visual and one worked real-world example. Context: Compare real and complex local transformations. End with a prediction or check-for-understanding.
Compare real and complex local transformations. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Complex plane, Complex exponential, Holomorphic functions and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Practical outcome: Evaluate an integral and explain each singularity.
Explain taylor expansions using a visual and one worked real-world example. Context: Evaluate an integral and explain each singularity. End with a prediction or check-for-understanding.
Explain laurent series using a visual and one worked real-world example. Context: Evaluate an integral and explain each singularity. End with a prediction or check-for-understanding.
Explain isolated singularities using a visual and one worked real-world example. Context: Evaluate an integral and explain each singularity. End with a prediction or check-for-understanding.
Explain residues using a visual and one worked real-world example. Context: Evaluate an integral and explain each singularity. End with a prediction or check-for-understanding.
Use the first concepts from this section. Evaluate an integral and explain each singularity. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain residue theorem using a visual and one worked real-world example. Context: Evaluate an integral and explain each singularity. End with a prediction or check-for-understanding.
Explain argument principle using a visual and one worked real-world example. Context: Evaluate an integral and explain each singularity. End with a prediction or check-for-understanding.
Explain rouche's theorem using a visual and one worked real-world example. Context: Evaluate an integral and explain each singularity. End with a prediction or check-for-understanding.
Explain conformal maps using a visual and one worked real-world example. Context: Evaluate an integral and explain each singularity. End with a prediction or check-for-understanding.
Evaluate an integral and explain each singularity. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Taylor expansions, Laurent series, Isolated singularities and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Practical outcome: Separate frequencies in a sampled vibration trace.
Explain orthogonal trigonometric series using a visual and one worked real-world example. Context: Separate frequencies in a sampled vibration trace. End with a prediction or check-for-understanding.
Explain fourier coefficients using a visual and one worked real-world example. Context: Separate frequencies in a sampled vibration trace. End with a prediction or check-for-understanding.
Explain convergence issues using a visual and one worked real-world example. Context: Separate frequencies in a sampled vibration trace. End with a prediction or check-for-understanding.
Explain fourier transform using a visual and one worked real-world example. Context: Separate frequencies in a sampled vibration trace. End with a prediction or check-for-understanding.
Use the first concepts from this section. Separate frequencies in a sampled vibration trace. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain inverse transform using a visual and one worked real-world example. Context: Separate frequencies in a sampled vibration trace. End with a prediction or check-for-understanding.
Explain convolution theorem using a visual and one worked real-world example. Context: Separate frequencies in a sampled vibration trace. End with a prediction or check-for-understanding.
Explain sampling theorem using a visual and one worked real-world example. Context: Separate frequencies in a sampled vibration trace. End with a prediction or check-for-understanding.
Explain aliasing using a visual and one worked real-world example. Context: Separate frequencies in a sampled vibration trace. End with a prediction or check-for-understanding.
Explain discrete fourier transform using a visual and one worked real-world example. Context: Separate frequencies in a sampled vibration trace. End with a prediction or check-for-understanding.
Separate frequencies in a sampled vibration trace. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Orthogonal trigonometric series, Fourier coefficients, Convergence issues and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Combine all three sections: Separate frequencies in a sampled vibration trace. Submit an original worked artifact, evidence or tests, assumptions and a reflection. Rubric: correctness, method, practical usefulness, reproducibility and communication. Self-reviewed, not certified.
6–12 balanced questions sampling every section of Complex Analysis and Fourier Methods, including interpretation, calculation or reasoning, a misconception and a new application. Passing is 70%; review errors before retrying.
Visual constructions, worked problems and proofs. Guided work leads to an independent attempt, with a diagram, explanation, trace, calculation or tested artifact. Question bodies are generated when opened in your private learning path. Programming submissions are stored as text; the platform does not execute submitted code.
Use original or appropriately licensed datasets. Practical science activities use models or supplied data unless a qualified supervisor has reviewed a physical experiment.
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