User:Selfworm/Math4

Take for example,


 * {| style="background:none;"


 * $$f(x, y)\,$$
 * $$= u(x, y) + iv(x, y)\,$$
 * $$= ln(z - a - ib)\,$$
 * $$= ln[(x - a)^2 + (y - b)^2]^{1/2} + i tan^{-1}({y - b\over x - a})\,$$
 * }
 * $$= ln(z - a - ib)\,$$
 * $$= ln[(x - a)^2 + (y - b)^2]^{1/2} + i tan^{-1}({y - b\over x - a})\,$$
 * }
 * $$= ln[(x - a)^2 + (y - b)^2]^{1/2} + i tan^{-1}({y - b\over x - a})\,$$
 * }

Note that,


 * {| style="background:none;"


 * $$u(z, 0) + iv(z, 0)\,$$
 * $$= ln[(z - a)^2 + b^2]^{1/2} + i tan^{-1}({b\over z - a})\,$$
 * $$= {1\over2}ln[(z - a)^2 + b^2] + {1\over2}ln[{(z - a) - ib\over (z - a) + ib}]\,$$
 * $$= ln (z - a - ib)\,$$
 * $$= f(x, y)\,$$
 * }
 * $$= ln (z - a - ib)\,$$
 * $$= f(x, y)\,$$
 * }
 * $$= f(x, y)\,$$
 * }
 * $$= f(x, y)\,$$
 * }