🕹️ Practice arcade

⚡ Polar Form Readiness Sprint

Modulus, argument, De Moivre's Theorem, and nth roots of complex numbers — the polar-form toolkit every later chapter leans on.

⚡ Sequences and Power Series Readiness Sprint

Sequence limits, the ratio test, geometric series, and a power series' own coefficients and center — the calc-II toolkit chapter 8 leans on.

🔢 Complex Arithmetic & Modulus Sprint

Multiply two complex numbers for the real part, find z times its own conjugate, or find |z| -- the three arithmetic facts everything else in this course leans on.

🌀 Argument, Growth & Limits Sprint

Multiply an argument by n under De Moivre's theorem, find |e^z| (the imaginary part never touches it), or evaluate a polynomial limit by straight substitution.

🔁 Contour Integral Theorems Sprint

Bound a contour integral with ML, evaluate one exactly with the Cauchy Integral Formula, or with the Residue Theorem -- three different guarantees about the same walk around a curve.

♾️ Series & Contour-Counting Sprint

Find a power series' radius of convergence, read a residue straight off a Laurent expansion, or turn a winding argument change into a zero-minus-pole count.

🗺️ Conformal Mapping & Special Functions Sprint

Find z^n's local scale factor, average a harmonic boundary function down to its center value, or climb the Gamma function with its own functional equation.

🌀 Complex Logarithm & Powers Sprint

Track a logarithm's imaginary part across branches, or find a real number raised to a purely imaginary power -- both land exactly where $e^{i\theta}$ says they should.

✅ Cauchy-Riemann & Differentiability Sprint

Solve for the missing coefficient that makes Cauchy-Riemann hold at a point, or find an entire polynomial's own derivative directly by the power rule -- two different routes to the same complex-differentiability question.

🎯 Identity & Maximum Principle Sprint

Use the Identity Theorem to force an entire function's value far from where it was defined, or find a linear harmonic function's own maximum over a closed disk.

📐 Continuity & Complex Trig Sprint

Patch a removable discontinuity with the right limit value, or split $\sin z$ into its real part via $\sin x\cosh y$ -- neither one needs anything beyond direct substitution.

🔁 Contour Integration Techniques Sprint

Parametrize a non-analytic integrand by hand around a circle, or lean on path independence and an antiderivative instead -- two genuinely different ways to evaluate a contour integral.

📈 Maximum Modulus & Taylor Coefficient Sprint

A boundary bound on $|f|$ that the Maximum Modulus Principle refuses to shrink for any interior point, or a Taylor coefficient's own derivative recovered by multiplying back in the missing $n!$.

🧮 Residue Calculus Toolkit Sprint

Read a pole-of-order-n residue off a polynomial's own leading coefficient, apply the simple-pole quotient shortcut, or find the residue at infinity from three finite ones -- three different ways into the same number.

🌊 Residue Applications & Rouche's Theorem Sprint

Close a semicircular contour to evaluate a real improper integral, or let Rouche's theorem hand you an exact zero count without ever locating a single zero.

🗺️ Mobius Maps & Riemann Sphere Sprint

Evaluate a bilinear transformation at a point, or project a point in the plane onto the Riemann sphere and read off its height.

🌡️ Conformal Mapping Applications Sprint

A wedge-shaped plate's steady-state temperature turns into a plain linear interpolation in the angle once the boundary condition is set up right.

🎯 Complex Plane Region Survival

Classify |z-z0| against r as the open disk, the closed disk, the circle itself, or the exterior -- one strict-vs-non-strict slip ends the round.

🧮 Conjugate Reciprocal Mastery

Use the conjugate to find the real part of $1/z$ — four in a row.

🎯 Geometric Series Check

A geometric distribution's own validity leans on this convergence test — get it right four times in a row.

🎯 Argument Quadrant Precision

A quadrant slip is the classic modulus/argument mistake — read a, b's signs correctly six times in a row.

🧭 Harmonic Conjugates & Analytic Continuation Mastery

Construct a harmonic conjugate's leading coefficient by hand, or track a branch point's monodromy around a multi-loop path -- four clean answers in a row, no clock.

🧭 Cauchy-Goursat & Liouville Mastery

An integral that vanishes because the integrand never sees a singularity inside the contour, or a bounded entire function that's secretly constant everywhere -- trust the theorem, not the numbers in front of you.

🔒 Schwarz Lemma & Schwarz-Pick Mastery

The sharpest possible modulus bound the Schwarz Lemma allows, or the sharpest possible derivative bound Schwarz-Pick allows -- five clean bounds in a row, no clock.