Difference between revisions of "Integral hyperbolic sine"
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The special function defined, for real $x$, by | The special function defined, for real $x$, by | ||
− | $$\operatorname{Shi}(x)=\int\limits_0^x\frac{\sinh t}{t}dt=i\operatorname{Si}(ix),$$ | + | $$\operatorname{Shi}(x)=\int\limits_0^x\frac{\sinh t}{t}\,dt=i\operatorname{Si}(ix),$$ |
where $\operatorname{Si}(x)$ is the [[Integral sine|integral sine]]. The integral hyperbolic sine can be represented by the series | where $\operatorname{Si}(x)$ is the [[Integral sine|integral sine]]. The integral hyperbolic sine can be represented by the series | ||
− | $$\operatorname{Shi}(x)=x+\frac{x^3}{3!3}+\frac{x^5}{5!5}+\ | + | $$\operatorname{Shi}(x)=x+\frac{x^3}{3!3}+\frac{x^5}{5!5}+\dotsb.$$ |
It is related to the [[Integral hyperbolic cosine|integral hyperbolic cosine]] $\operatorname{Chi}(x)$ by | It is related to the [[Integral hyperbolic cosine|integral hyperbolic cosine]] $\operatorname{Chi}(x)$ by |
Latest revision as of 20:40, 1 January 2019
The special function defined, for real $x$, by
$$\operatorname{Shi}(x)=\int\limits_0^x\frac{\sinh t}{t}\,dt=i\operatorname{Si}(ix),$$
where $\operatorname{Si}(x)$ is the integral sine. The integral hyperbolic sine can be represented by the series
$$\operatorname{Shi}(x)=x+\frac{x^3}{3!3}+\frac{x^5}{5!5}+\dotsb.$$
It is related to the integral hyperbolic cosine $\operatorname{Chi}(x)$ by
$$\operatorname{Chi}(x)+\operatorname{Shi}(x)=\operatorname{Li}(e^x),$$
where $\operatorname{Li}$ is the integral logarithm.
Sometimes it is denoted by $\operatorname{shi}(x)$.
For references see Integral cosine.
Comments
This function, which is seldom used because of its relation with the sine integral, is also called the hyperbolic sine integral. It can, of course, be defined (as above) for $z\in\mathbf C$.
Integral hyperbolic sine. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_hyperbolic_sine&oldid=43638