Euler Sine Canonical Product

Worked Example

Let us calculate the partial product for z=0.5 with a product depth of N=4.

1. The formula is prod_n=1^4 (1 - z^2/n^2).
2. For n=1: 1 - (0.5)^2/1^2 = 1 - 0.25 = 0.75
3. For n=2: 1 - (0.5)^2/2^2 = 1 - 0.25/4 = 0.9375
4. For n=3: 1 - (0.5)^2/3^2 = 1 - 0.25/9 = 0.972222...
5. For n=4: 1 - (0.5)^2/4^2 = 1 - 0.25/16 = 0.984375
6. Partial product real: 0.75 * 0.9375 * 0.972222... * 0.984375 = 0.67291259765625
7. The sine quotient real is exactly sin(pi*0.5)/(pi*0.5) = 1/(pi/2) = 2/pi = 0.6366197723675814.
8. Absolute complex error is |0.67291259765625 - 0.6366197723675814| = 0.03629282528866862.

Interactive Model

You can interact with the interactive model. Follow these steps to observe the convergence:

1. Set Real coordinate to 0.5.
2. Set Imaginary coordinate to 0.
3. Set Product depth to 4 and observe the Absolute complex error.
4. Increase the Product depth to 16 and observe that the error decreases.
5. Change the Imaginary coordinate to 0.5 and notice how the complex trajectory changes.

Limitations

The finite product is not exact away from its represented zeros, and the complex error may not decrease monotonically in all regions. Additionally, this represents the quotient sin(pi*z)/(pi*z), which has a removable singularity with value 1 at z=0. The origin itself is NOT a zero of this factorization.

References

For more information, see the following primary sources:

• DLMF 4.22.E1 Euler Infinite Product for Sine
• Mathlib EulerSineProd