Digamma Time

Calculus Level 2

ψ ( 1 2 + 100000000 ) − ψ ( 1 2 − 100000000 ) = ? \psi\left(\frac 12+100000000 \right)-\psi \left(\frac 12-100000000 \right)= \ ?

Bonus: Solve for any positive integer n n instead of 100000000 100000000 .

Notation: ψ ( ⋅ ) \psi (\cdot) denotes the digamma function .


The answer is 0.

Aruna Yumlembam by Aruna Yumlembam , 16, India

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1 solution

Chew-Seong Cheong
Jul 12, 2020

X = ψ ( 1 2 + n ) − ψ ( 1 2 − n ) where n is an integer. = ψ ( 1 2 + n ) − ψ ( 1 − ( 1 2 + n ) ) As ψ ( z ) − ψ ( 1 − z ) = − π cot ⁡ ( π z ) = − π cot ⁡ ( ( n + 1 2 ) π ) = 0 \begin{aligned} X & = \psi \left(\frac 12 + n \right) - \psi \left(\frac 12 - n \right) & \small \blue{\text{where }n \text{ is an integer.}} \\ & = \psi \left(\frac 12 + n \right) - \psi \left(1 - \left(\frac 12 + n \right) \right) & \small \blue{\text{As }\psi (z) - \psi(1-z) = - \pi \cot (\pi z)} \\ & = - \pi \cot \left(\left(n+\frac 12\right)\pi \right) \\ & = \boxed 0 \end{aligned}

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