Singularities

List of finite values corresponding to the singularities (poles) of well-known functions.

$H_n$ denotes the n-th harmonic number (with $H_0 = 0$ ).

$G_n$ denotes the n-th Gregory number or coefficient.

$\gamma = \zeta(1) \approx 0.57721566490153286\ldots$ denotes the Euler-Mascheroni constant.

Poles of Order 0 (Removeable Singularities)

Function sinc( )

The unnormalized and normalized function $sinc(z) \; (z\in\mathbb{C})$ has one removable singularity at $z=0$ . Its finite value at its pole $z=0$ is:

$sinc(0) = \dfrac{\sin(0)}{0} = 1 \qquad \,$ (unnormalized)

$sinc(0) = \dfrac{\sin(0\pi)}{0\pi} = 1 \qquad \,$ (normalized)

Tangent function tan( )

The tangent function $\tan(z) \; (z\in\mathbb{C})$ has removable singularities at every odd multiple of $~\dfrac{\pi}{2}$ . Its finite value at its poles is:

$\tan\left(\dfrac{(2n+1)\pi}{2}\right) = -\dfrac{1}{2\pi} \qquad (n\in\mathbb{N})$

$\tan\left(-\dfrac{(2n+1)\pi}{2}\right) = -\tan\left(\dfrac{(2n+1)\pi}{2}\right) = \dfrac{1}{2\pi} \qquad (n\in\mathbb{N})$

In particular, $\tan\left(\dfrac{\pi}{2}\right) = -\dfrac{1}{2\pi}$ .

Cotangent function cot( )

The cotangent function $\cot(z) \; (z\in\mathbb{C})$ has removable singularities at every multiple of $\pi$ . Its finite value at its poles is:

$\cot\left(\dfrac{\pi n}{2}\right) = 0 \qquad (n\in\mathbb{Z})$

In particular, $\cot(\pi) = \cot(0) = 0$ .

Natural log of the sine function ln(sin( ))

The natural log of sine function $\ln(\sin z) \; (z\in\mathbb{C})$ has removable singularities at every multiple of $\pi$ . Its finite value at its poles is:

$\ln(\sin \pi n) = -(\gamma+\ln2)$

Natural log of the cosine function ln(cos( ))

The natural log of cosine function $\ln(\cos z) \; (z\in\mathbb{C})$ has removable singularities at every odd multiple of $\dfrac{\pi}{2}$ . Its finite value at its poles is:

$\ln\left(\cos\pi\left(n+\dfrac{1}{2}\right)\right) = -(\gamma+\ln2)$

Natural log of the tangent function ln(tan( ))

The natural log of the tangent function $\ln(\tan z) \; (z\in\mathbb{C})$ has removable singularities at every odd multiple of $\dfrac{\pi}{2}$ . Its finite value at its poles is:

$\ln\left(\tan\pi\left(n+\dfrac{1}{2}\right)\right) = \gamma+\ln2$

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