Euler's Formula — Power Series
Interactive decomposition of Figure 1.9 from Visual Complex Analysis (Tristan Needham, Oxford University Press, 2023), Section 1.2.3. The power series produces a spiral in the complex plane where each term is rotated 90° from the previous one. As more terms are added, the partial sum converges to the point on the unit circle.
Partial Sums Building the Spiral
Each figure shows the partial sum for θ = π/3 (60°). Vectors are drawn head-to-tail: term 0 (real, along Re axis), term 1 (rotated 90°), term 2 (rotated another 90°), and so on. The blue dot is the partial sum endpoint; the green dashed circle marks the target . The dashed blue line from the origin to the partial sum endpoint is the resultant vector.
Interactive: Adjust θ and Number of Terms
Drag the θ slider to change the angle. Increase the number of terms to watch the spiral converge toward the unit circle. The blue arrow shows the current partial sum; the green dot marks the target .
The Argument
Step 1. The exponential function can be defined by its power series: , which converges for all real x.
Step 2. Substitute x = iθ into the series. Since i2 = -1, each successive term is rotated by 90° relative to the previous one, producing a spiral in the complex plane: .
Step 3. Split into real and imaginary parts: , where and . These are the power series for cos θ and sin θ.
Step 4. Without invoking Taylor's Theorem, we can show |eiθ| = 1 by noting that d/dθ (C2 + S2) = 0 (using C' = -S and S' = C), so the magnitude is constant at 1.
Step 5. The angle Θ(θ) of eiθ satisfies dΘ/dθ = 1, so Θ = θ. The spiral converges to the point on the unit circle at angle θ: .
Reference: Needham, Tristan. Visual Complex Analysis, Section 1.2.3 "Power Series Argument", Figure 1.9. Oxford University Press, 2023. 25th Anniversary Edition.