Formula:5624

Digital Library of Mathematical Functions ID 14.20.E6

${\displaystyle{\displaystyle P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{ie^{-\mu\pi i}}{\sinh\left(\tau\pi\right)\left|\Gamma\left(\mu+\frac{1}{2}+i\tau\right)\right|^{2}}\*\left(Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)-Q^{\mu}_{-\frac{1}{2}-i\tau}\left(x\right)\right),}}$

Constraint(s)

${\displaystyle{\displaystyle\tau\neq 0}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}$: associated Legendre function of the first kind
• ${\displaystyle{\displaystyle Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}$:
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\sinh\NVar{z}}}$: hyperbolic sine function
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle\tau}}$: real variable
• ${\displaystyle{\displaystyle\mu}}$: general order
 

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