View source for Tangent

The trigonometric function

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another notation is: tg. Its domain of definition is the entire number axis with the exception of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092090/t0920902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092090/t0920903.png" />. The tangent is an unbounded, odd and periodic (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092090/t0920904.png" /> as the smallest positive period) function. The tangent and the [[Cotangent|cotangent]] are connected by the relation

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The inverse function to the tangent is called the arctangent.

The derivative of the tangent is:

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The indefinite integral of the tangent is:

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The tangent has a series expansion:

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The tangent of a complex argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092090/t0920909.png" /> is a meromorphic function with zeros at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092090/t09209010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092090/t09209011.png" />.



====Comments====
The general term in the series expansion of the tangent is:

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092090/t09209013.png" /> are the [[Bernoulli numbers|Bernoulli numbers]].

See also [[Trigonometric functions|Trigonometric functions]].

The addition formula of the tangent is:

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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Abramowitz,   I.A. Stegun,   "Handbook of mathematical functions" , Dover, reprint  (1965)  pp. 71ff</TD></TR></table>