A mapping from a complex plane z to a complex plane w such that
w = f(z) and f'(z_o) ≠ 0 (f'(z_o) is the derivative of f at the point z_o).
f(z) is conformal everywhere in the z-plane If f'(z) ≠ 0 everywhere in the z-plane. For example, the mapping f(z) = e^z is conformal everywhere in the z-plane since f'(z) = e^z ≠ 0 everywhere.

A plane in which the points are complex numbers. For example, a complex z-plane has the complex number z = x + yi as its points, where (x,y) is the cartesian point and i² = -1.

A function that evaluates to a complex number when its argument is a real number.

The theorem that equates the definite integral of a function to the product of the interval over which the function is integrated and the value of the fuction at a point within the interval.

The theorem states that if the function f(x) is continuous on the interval [a b], then there exists at least one number ξ, in [a b] that satisfies the following equation:

∫^a_bf(x)dx = f(ξ)(b-a)

The summation of a continuous function between known boundary values. For example,∫^b_af(x)dx is a definite integral of f(x) over the closed interval [a b].

A very important operator in mathematical physics. Basically, it is the summation of the second derivatives of a function with respect to each of its independent variables. Now suppose we have a function u(x,y) in two dimensions, the Laplacian Operator is expressed as follows:

∇² = δ²/δx² + δ²/δy².

For the function u(x, y, z) in three dimensions, the Laplacian Operator is expressed as follows:

A function of a function.

Consider the following second-order linear partial differential equation (PDE) in two variables:

Au_xx + Bu_xy + Cu_yy + Du _x + Eu_y + Fu = G(x,y) -----(Eqn 1)

Where, A, B, C, D, E, F and G can be constants or functions of x and y.

The above PDE (Eqn 1) is elliptic if B² - 4AC < 0. Elliptic PDE describe steady-state phenomena.

Consider the following second-order linear partial differential equation(PDE)in two varibles:

Au_xx+ Bu_xy + Cu _yy + Du_x + Eu_y + Fu = G(x,y) ----(Eqn 1)

Where A, B, C, D, E, F and G can be constants or functions of x and y.

The above PDE (eqn 1) is hyperbolic if B² - 4AC > 0. Hyperbolic PDE describe wave motion and vibrating systems.

Consider the following second-order linear partial differential equation (PDE) in two variables:

Au_xx + Bu_xy + Cu_yy + Du_x + Eu_y + Fu = G(x,y)----(eqn 1)

Where A, B, C, D, E, F and G can be constants or functions of x and y.

The above PDE (eqn 1) is parabolic if B² - 4AC = 0. Parabolic PDE describe heat flow and diffusion processes.