App Complex Analysis - Android

ivan gabrovski

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About this app

- **Complex calculator** expressions in z; algebraic, polar and exponential forms, |w|, arg, conjugate, 1/w,
- **Domain coloring** phase + modulus, phase only, Re/Im grid; drag, pinch, double-tap to zoom, tap to read f(z).
- **Roots and zeros** polynomial roots (Aberth–Ehrlich + multiplicities, Vieta check); zeros of any f(z) in |z| < R (Newton) cross-checked with the argument principle.
- **Analysis** Cauchy Riemann and Wirtinger derivatives at z₀, contour integral on |z − c| = r, residue sum, Z − P, Laurent coefficients a₋₅…a₅; 10 built-in examples.
- **Saved examples** – on Domain coloring, Roots and Analysis the *Examples* list holds your own saved inputs (★, newest first) above the built-in ones; *Save current* stores the inputs (for plots also the color mode and visible region), long-press a ★ entry to delete it. Stored locally in
## Expression syntax
`+ - * / ^`, implicit multiplication (`2z`, `3i`, `(z-1)(z+1)`), constants `i pi π e`,
functions `sin cos tan sinh cosh tanh exp log ln sqrt asin acos atan conj re im abs arg`.
The images are calculated using the same formulas as ColorMap.java in the application.
The idea. The graph of f: ℂ → ℂ is four-dimensional and cannot be drawn directly. Therefore, each point z in the plane is colored according to the value w = f(z):
hue = arg w. Red is the positive real axis (arg 0), then yellow → green → cyan (±π) → blue → magenta → back to red;
brightness = |w|. Used only in module modes.
The benchmark is f(z) = z (color_key.png): there the color of each point is its own argument.
How to read zeros and poles. Around a zero or pole all the colors come together at one point. The direction shows which is:
zero of row n: counterclockwise the colors go red → yellow → green → blue, n times;
pole of order n: the same, but in reverse order;
the order is the number of complete revolutions of the color wheel. In domain_coloring_modes.png at z = 2 + i (zero of order 2) each color appears twice. At ±1 (simple zeros) it appears once at both poles.
This is a direct visualization of the argument principle. Look around any closed curve and count how many times the colors go around the wheel forward minus backward. The result is Z − P inside it.
The three modes:
Phase + modulus. The brightness increases with log₂|f| and drops sharply every time |f| doubles. Concentric stripes are obtained, which thicken towards the zeros and poles. Towards the zeros the dark edge is on the inside of the stripe, towards the poles it is the other way around. In addition, there are 12 weak radial sectors along the phase. Their intersecting lines form a “lattice” that is conformal: f preserves angles, so the lines intersect at right angles wherever f′ ≠ 0.
Phase only. The cleanest picture for counting zeros and poles, without modulus information. Convenient for comparing functions with the same zero structure.
Re/Im lattice. Dark lines where Re f or Im f is an integer. These are the inverse images of the square lattice from the w-plane. The squares remain square (conformity), but their size is ~1/|f′|. Therefore, around the poles the lattice is dense, and near zero at z = 2 + i it is sparse. The lines there intersect at 45° instead of 90°, because f′ = 0 at zero of order 2 and the conformality is violated.
Special colors:
white: f is infinite (exactly at the pole or at overflow);
gray: f is undefined (NaN), for example 0/0;
black: exact zero.
Useful experiments in the app:
exp(1/z): a real singularity is visible as it approaches 0. All colors repeat infinitely many times in each neighborhood (Cazarati–Weierstrass theorem).
sqrt(z) and log(z): sharp jump in color on the negative real axis. This is the intersection of the main branch, not a feature of the function itself.
sin(z)/z: there is nothing special at z = 0 because the singularity is removable.
conj(z): the colors rotate in the opposite direction as at a pole, but without infinity. This is what a non-holomorphic function looks like.