CONFORMAL MAPPINGS 1 (Complex Analysis)

Apply translation in the complex plane with w = z + c to shift every point by the complex constant c, preserving shape and size.

Explore conformal mappings by computing the coefficient of magnification and angle of rotation for w = f(z), using the jacobian and Cauchy–Riemann relations; illustrated with z^2.

Rotate z-plane by gamma with w = e^{i gamma} z, preserve modulus and shift angle; gamma = pi/4 maps the triangle to u = ± v and v = 1/√2.

Magnification is a homothetic transformation with positive factor c, stretching for c>1 or contracting for 0<c<1; in the example w=2z, the triangle maps to u+v=2.

this lecture covers rotation and magnification as conformal transformations, using w = beta z in polar form, and demonstrates a 45-degree anticlockwise rotation with magnification.

Explore the inversion w = 1/z in polar form, mapping r e^{i theta} to (1/r) e^{-i theta}, sending the origin to infinity and points on the unit circle to themselves.

The infinite strip y ≥ 1/4 and y ≤ 1/2 maps under w = 1/z to the region between the circles u^2+(v+2)^2=4 and u^2+(v+1)^2=1.

Using the map w=(2z+3)/(z-4), the lecture derives z=(4w+3)/(w-2) and shows the circle x^2+y^2-4x=0 maps to the straight line 4u+3=0 in the w-plane.

The transformation w = z^2 maps the circle |z−1|=1 to a cardioid in the w plane, with rho = 2(1 + cos phi) and phi = 2 theta.

Explore linear transformations mapping a z-plane rectangle to a w-plane region via w = alpha z + beta, magnify by |alpha|, rotate by arg(alpha), and translate by |beta|; alpha=1+i, beta=2-i.

Study bilinear transformations with nonzero determinant ad − bc, their inverse, and exceptional points, establishing a 1-to-1 conformal mapping between the extended complex planes.

Explore a solved example of bilinear transformation and inversion transformation, deriving t1 inverse and t2 inverse, and evaluating the products T2T1Z and T1T2Z.

Derive the bilinear transformation that maps z1=2, z2=i, z3=-2 to w1=1, w2=i, w3=-1, yielding w = (3 z + 2 i) / (i z + 6).

Derive the bilinear transformation mapping z1=2, z2=i, z3=-2 to w1=1, w2=i, w3=-1 using an alternative method, solving for a, b, c, d in w=(a z+b)/(c z+d).

Compute the bilinear transformation mapping z points 0, iota, -iota to w points 1, -1, 0, yielding w = (z + iota)/(iota − 3z) as the required mapping.

Derive the bilinear transformation w = i z using the cross-ratio method to map z = 1, 0, -1 to w = i, 0, -i in complex analysis.

Derive the bilinear transformation mapping z1=infinity, z2=i, z3=0 to w1=0, w2=i, w3=infinity using the cross ratio, yielding w = -1/z.

Explore how Möbius transformations map circles or straight lines to circles or straight lines, using cross ratios to establish preservation under bilinear transformations.

Bilinear transformations map circles or lines to circles or lines and inverse points to inverse points, with Steiner circles illustrating how z1 and z2 map to w1 and w2.

Solve cz^2+(d−a)z−b=0 to find fixed points of w=(az+b)/(cz+d); if c≠0, two or one fixed point depending on the discriminant; if c=0, z=b/(d−a) and infinity.

Derive the normal form of a bilinear transformation with fixed points alpha and beta, yielding w−alpha over w−beta = lambda (z−alpha)/(z−beta) where lambda = (d+c alpha)/(d+c beta).

Derive that a bilinear transformation with a single fixed point alpha can be written as 1/(w - alpha) = 1/(z - alpha) + lambda; alpha = (a - d)/(2c), lambda = c/(d + c alpha).

Determine bilinear transformations from fixed points: with one finite and one infinite fixed point, c=0 and a≠d give w−α=(a/d)(z−α); with only infinite fixed point, a=d yields a translation.

Explore conformal mappings of complex planes via elliptic, hyperbolic, parabolic and loxodromic bilinear transformations, analyzing fixed points, normal forms, and the role of lambda.

The bilinear transformation w equals z^2 minus z has fixed points z equals 0 and z equals 1, and the normal form yields w/(w-1) = (1/2)·z/(z-1), a hyperbolic map.

Identify the fixed point z = 2 for the bilinear transformation w = (3z-4)/(z-1) and express its parabolic normal form as 1/(w-2) = 1/(z-2) + 1, with lambda = 1.

Identify the fixed points of the bilinear transformation w=(z-1)/(z+1), derive its normal form, and show that the mod of lambda equals one, yielding an elliptical transformation.