Complex analysis

• What Möbius Map Rotates the Riemann Sphere About the x-Axis?
  With the standard stereographic coordinate z = (X + iY)/(1 − Z), rotation by θ about the positive x-axis is a Möbius map using θ/2.
• How Do Dogbone Contours, Branch Cuts, and Residues at Infinity Work?
  Learn how a dogbone contour follows both sides of a branch cut, how to track orientation, and when the residue at infinity applies.
• Why the Minimum Modulus Principle Follows from the Maximum Modulus Principle
  The minimum modulus principle follows by applying the maximum modulus principle to 1/f near any point where a holomorphic function is nonzero.
• Can Composition Increase a Power Series’ Radius of Convergence?
  Yes. Composition can cancel singularities of the inner function, so the resulting Taylor series may have a larger radius of convergence.
• How to Find the Real and Imaginary Parts of a Polynomial
  Find a polynomial’s real and imaginary parts by substituting z = x + iy, expanding, and collecting terms; verify the result with SymPy.