We begin by stating, without proof, that the equation for induced drag `C`_{Di} over a wing with circulation Γ(`y`) in a flow with density `ρ`_{∞} and a downwash velocity of `w`(`y`). `y` represents position on the wing measured from the root, varying from -`b`/2 to +`b`/2, where b is wingspan.

`D`_{i} = -∫_{-b/2}^{+b/2}`ρ`_{∞}`w`(`y`)Γ(`y`)`dy`

Next, we take the elliptical distribution of circulation around a wing, which depends on `y`:

Γ(`y`) = Γ_{0}√(1 - (`y`/(`b`/2))^{2})

Finally, the downwash velocity for this planform simplifies to:

`w`(`y`) = -Γ_{0}/(2`b`)

We may now substitute into the integral expression:

`D`_{i} = -∫_{-b/2}^{+b/2}`ρ`_{∞}(-Γ_{0}/(2`b`))(Γ_{0}√(1 - (`y`/(`b`/2))^{2}))

We introduce the coordinate transfomation `y` = -`s` cos `φ`, and simplify the expression to:

`D`_{i} = (`ρ`_{∞}Γ_{0}^{2}/(2b))∫_{0}^{π}(`b`/2) sin `φ`√(1 - cos^{2} `φ`) `dφ`

`D`_{i} = (`π`/8)`ρ`_{∞}Γ_{0}^{2}

We now apply the formula for the drag coefficient:

`C`_{Di} = `D`_{i}/(0.5*`ρ`_{∞}`V`_{∞}^{2}`S`)

`C`_{Di} = `π`Γ_{0}^{2}/(4`V`_{∞}^{2}`S`)

This is all well and good, but the circulation at the wing root is not a practical design variable. We then state, without proof, that based on the expression for the lift coefficient of the same wing:

Γ_{0} = 2`C`_{L}`V`_{∞}`S`/(`πb`)

Thus, we can substitute into our expression for the drag coefficient:

`C`_{Di} = {`π`/(4`V`_{∞}^{2}`S`)}{2`C`_{L}`V`_{∞}`S`/(`πb`)}^{2}

`C`_{Di} = `C`_{L}^{2} / (`π`*`AR`)

We have now derived the coefficient of induced drag for a wing with an elliptical circulation, and thus, elliptical lift distribution and planform (these follow directly from the Kutta-Joukowski lift theorem and the proportionality of chord to circulation).

Now, *poof* I present the formula for the coefficient of induced drag in general:

`C`_{Di} = `C`_{L}^{2} / (`π`*`AR`*`e`)

where the airplane efficiency factor `e` ≤ 1. Note then that for the elliptical planform, `e` = 1. Therefore, a wing with an elliptical planform is the most efficient (provided aspect ratio and lift coefficient are fixed), because it has the least possible value for its coefficient of induced drag.