FT5: Formulas using Complex Hyperbolic Functions

This is the fifth in a series of posts on Fibonacci trigonometry:

Part 1: Basic Definitions

Part 2: The Canonical Basis and Closed-Form Expressions

Part 3: The Basic Analogies

Part 4: Fibonacci Formulas Using sinh, etc. — Real Number Version

Part 5: Fibonacci Formulas Using sinh, etc. — Complex Number Version

Part 6: Trigonometric-Style Identities

Part 7: The Fibonacci Tangent and Cotangent Functions

Part 8: Recurrence Relations for Ftan and Fcot

 

In part 4, we saw formulas for the Fibonacci numbers and related sequences in terms of the hyperbolic functions. Here we'll see how, by using complex numbers, we can avoid splitting into even and odd cases.

Let ψ = ln φ + πi/2. (Anything that differs from this by an integer multiple of 2πi will also work.)

Then:

• F(n) = 2 sinh(nψ) / (√5 iⁿ)
• L(n) = 2 cosh(nψ) / iⁿ

• Ftan(n) = tanh(nψ) / √5 
• Fcot(n) = √5 coth(nψ) 

• Fsec(n) = iⁿ sech(nψ) / 2
• Fcsc(n) = √5 iⁿ csch(nψ) / 2