Asian Journal of Mathematics and Computer Research

Suppose that a simple closed curve C is defined parametrically in the complex w-plane

                                                        u = u (θ) , v = v (θ) , 0 ≤ θ ≤ 2π

Then the area A enclosed by C is given by,

                                                      \(A=\frac{1}{2} \int_0^{2 \pi}\left(u \frac{d v}{d \theta}-v \frac{d u}{d \theta}\right) d \theta\)

If there is a conformal mapping Ψ_r,from the exterior of D_r ,r > 1 to the exterior of C_r of the form

                                                    \(\Psi_r(z)=z+\sum_{n=0}^{\infty} \frac{b_n}{z^n}\)

then by Gronwall’s area formula, the area A_r of the region B_r enclosed by C_r is given by,

                                                  \(A_r=\pi\left(r^2-\sum_{n=1}^{\infty} n\left|b_n\right|^2 r^{-2 n}\right)\)

We use Gronwall’s area formula to find the area of some differents regions as circles with radius r centred ot the origin lying in the complex w-plane , ellipses and lemniscates A lemniscate shapped with any number of leaves .The boundary of m-leafed symmetric lemniscate is the set

                                                   \(\left\{w \in \mathbb{C},\left|w^m-1\right|=0\right\}, m=2,3,4 \ldots . .\)

We use Laurent and Taylor series expansions of conformal mapping from the exterior of the unit disk to either of these regions to compute the area of them.