Phase Portrait Lab

Phase Portrait Labcomplex functions, mapped and colored

Function

f(z) =

u(x,y) =

v(x,y) =

degree 4 poles 2 rewrites the formula above

Nothing here yet — examples appear as they debut in class.

Parameters

Any letter your formula uses other than z becomes a parameter — a, b, c, d, or a subscripted one such as a₃, r₁, or p₁ (r and p need their subscript). Drag it in the thumbnail, type it, or press ▶ to swing it in a circle. z and w also place its handle on that plane, so you can drag the number where it actually sits; both at once is allowed.

m, n and k are different: they are integers, stepped on a slider instead of dragged, so z^(m/n) and a branch counter are typeable. Their subscripts (m₁, n₂, …) are integers too. Click a slider's range to change its bounds.

In u+iv mode every parameter is real, on a slider. It has to be: u and v are real functions of x and y, so a complex a inside u would make u complex and the two labels would be lying. Switch back to f(z) and the parameter is complex again, where you left it.

Coloring

Phase portrait (input plane, by f)

Modulus contours (|f| ×2 each ring) Phase contours (12 sectors)

opacity100

Color output plane (reference)

opacity100

show: grid axes labels

White background (for screenshots printed on paper)

Shapes

Snap points to grid (handles only) Disc annulus & sector too — drag its handles

outer r1

inner r0

start θ0π

end θ2π

circles3

spokes8

Grid drag its handles

size×

lines ∥13

lines ⊥13

cell×□

right angle lock cell

Draw sketch on either plane

Image…

Images can also be dropped onto either plane, or pasted (⌘V / Ctrl+V).

Hue is the phase φ(f(z)); arg values use the branch (−π, π].
Complex Analysis · Block 1 · Proof School

z sphere

drag to orbit · scroll to zoom

sphere centre R = 0.500

|f| over z

drag to orbit · scroll to zoom

1.00×

input z

scroll to zoom · drag to pan · double-click drops the test point · alt-drag pans while a draw tool is armed

w sphere

drag to orbit · scroll to zoom

sphere centre R = 0.500

output w

branches