Sine integral - Revision history

Vipul: /* First antiderivative */

2011-12-19T19:24:03Z

First antiderivative

← Older revision		Revision as of 19:24, 19 December 2011
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	===First antiderivative===		===First antiderivative===

	{{'''WHAT WE USE''': [[integration by parts]]}}		{{quotation\|'''WHAT WE USE''': [[integration by parts]]}}

	We have:		We have:

Vipul: /* Key data */

2011-12-19T19:23:41Z

Key data

← Older revision		Revision as of 19:23, 19 December 2011
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	\| [[power series]] and [[Taylor series]] \|\| <math>\sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{(2k + 1)(2k + 1)!}</math>. First few terms:<br><math>x - \frac{x^3}{3 \cdot 3!} + \frac{x^5}{5 \cdot 5!} - \dots</math>		\| [[power series]] and [[Taylor series]] \|\| <math>\sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{(2k + 1)(2k + 1)!}</math>. First few terms:<br><math>x - \frac{x^3}{3 \cdot 3!} + \frac{x^5}{5 \cdot 5!} - \dots</math>
	\|}		\|}

			==Differentiation==

			===First derivative===

			The first derivative is the [[sinc function]]. This follows from the way the sine integral is defined in the first place.

			===Second derivative===

			See [[sinc function#First derivative]].

			==Integration==

			===First antiderivative===

			{{'''WHAT WE USE''': [[integration by parts]]}}

			We have:

			<math>\int \operatorname{Si}(x) \, dx = x \operatorname{Si}(x) + \cos x + C</math>

			We obtain this using integration by parts as follows. We take <math>1 \, dx</math> as the part to integrate, getting:

			<math>\operatorname{Si}(x) \cdot x - \int \frac{\sin x}{x} \cdot x \, dx</math>

			This simplifies to the desired answer.

Vipul: /* Key data */

2011-12-19T19:20:58Z

Key data

← Older revision		Revision as of 19:20, 19 December 2011
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	\| [[antiderivative]] \|\| <math>x \operatorname{Si}(x) + \cos x + C</math>		\| [[antiderivative]] \|\| <math>x \operatorname{Si}(x) + \cos x + C</math>
	\|-		\|-
	\| [[power series]] and [[Taylor series]] \|\| <math>\sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}(2k + 1)(2k + 1)!}</math>. First few terms:<br><math>x - \frac{x^3}{3 \cdot 3!} + \frac{x^5}{5 \cdot 5!} - \dots</math>		\| [[power series]] and [[Taylor series]] \|\| <math>\sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{(2k + 1)(2k + 1)!}</math>. First few terms:<br><math>x - \frac{x^3}{3 \cdot 3!} + \frac{x^5}{5 \cdot 5!} - \dots</math>
	\|}		\|}

Vipul at 19:20, 19 December 2011

2011-12-19T19:20:31Z

← Older revision		Revision as of 19:20, 19 December 2011
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	<math>\operatorname{sinc} \, x := \left\lbrace \begin{array}{rl} 1, & x = 0 \\ \frac{\sin x}{x} , & x \ne 0 \\\end{array}\right.</math>		<math>\operatorname{sinc} \, x := \left\lbrace \begin{array}{rl} 1, & x = 0 \\ \frac{\sin x}{x} , & x \ne 0 \\\end{array}\right.</math>

			==Key data==

			{\| class="sortable" border="1"
			! Item !! Value
			\|-
			\| default [[domain]] \|\| all [[real number]]s, i.e., all of <matH>\R</math>
			\|-
			\| [[range]] \|\| <math>[-\alpha,\alpha]</math> where <math>\alpha \approx 1.8519</math>.<br>[[absolute maximum value]]:<math>\alpha</math>, [[absolute minimum value]]: <math>-\alpha</math>
			\|-
			\| [[period]] \|\| none, the function is not periodic
			\|-
			\| [[horizontal asymptote]]s \|\| <math>y = \pi/2</math> as <math>x \to +\infty</math> and <math>y = -\pi/2</math> as <math>x \to -\infty</math>.
			\|-
			\| [[local maximum value]]s and points of attainment \|\| local maximum values are attained at all odd positive integer multiples of <math>\pi</math> and even negative integer multiples of <math>\pi</math>. The values differ at each point.
			\|-
			\| [[local minimum value]]s and points of attainment \|\| local minimum values are attained at all even positive integer multiples of <math>\pi</math> and odd negative integer multiples of <math>\pi</math>. The values differ at each point.
			\|-
			\| [[point of inflection\|points of inflection]] \|\| at <math>(0,0)</math> and at points whose <math>x</math>-coordinate is close to an odd multiple of <math>\pi/2</math>, except near <math>\pm \pi/2</math>.
			\|-
			\| important symmetries \|\| the function is an [[odd function]]
			\|-
			\| [[derivative]]] \|\| [[sinc function]] <math>x \mapsto (\sin x)/x, x \ne 0; 1, x = 0</math>
			\|-
			\| [[second derivative]] \|\| <math>(x \cos x - \sin x)/x^2</math> for <math>x \ne 0</math>, <math>0</math> for <math>x = 0</math>
			\|-
			\| [[antiderivative]] \|\| <math>x \operatorname{Si}(x) + \cos x + C</math>
			\|-
			\| [[power series]] and [[Taylor series]] \|\| <math>\sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}(2k + 1)(2k + 1)!}</math>. First few terms:<br><math>x - \frac{x^3}{3 \cdot 3!} + \frac{x^5}{5 \cdot 5!} - \dots</math>
			\|}

Vipul: /* Definition */

2011-09-04T13:23:35Z

Definition

← Older revision		Revision as of 13:23, 4 September 2011
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	<math>\operatorname{Si}(x) := \int_0^x \operatorname{sinc} \, t \, dt</math>		<math>\operatorname{Si}(x) := \int_0^x \operatorname{sinc} \, t \, dt</math>

	Recall that the sinc function is defined as follows:		Recall that the sinc function is defined in terms of the [[defining ingredient::sine function]] as follows:

	<math>\operatorname{sinc} \, x := \left\lbrace \begin{array}{rl} 1, & x = 0 \\ \frac{\sin x}{x} , & x \ne 0 \\\end{array}\right.</math>		<math>\operatorname{sinc} \, x := \left\lbrace \begin{array}{rl} 1, & x = 0 \\ \frac{\sin x}{x} , & x \ne 0 \\\end{array}\right.</math>

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== The function, denoted $\operatorname{Si}$ , is defined as the [[defining ingredient::def..."

2011-09-04T13:22:51Z

Created page with "{{particular function}} {{angular function radian convention}} ==Definition== The function, denoted <math>\operatorname{Si}</math>, is defined as the [[defining ingredient::def..."

New page

{{particular function}}
{{angular function radian convention}}

==Definition==

The function, denoted <math>\operatorname{Si}</math>, is defined as the [[defining ingredient::definite integral]] of the [[defining ingredient::sinc function]] from 0 to the input point:

<math>\operatorname{Si}(x) := \int_0^x \operatorname{sinc} \, t \, dt</math>

Recall that the sinc function is defined as follows:

<math>\operatorname{sinc} \, x := \left\lbrace \begin{array}{rl} 1, & x = 0 \\ \frac{\sin x}{x} , & x \ne 0 \\\end{array}\right.</math>

Sine integral - Revision history

Vipul: /* First antiderivative */

Vipul: /* Key data */

Vipul: /* Key data */

Vipul at 19:20, 19 December 2011

Vipul: /* Definition */

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== The function, denoted \operatorname{Si}, is defined as the [[defining ingredient::def..."

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== The function, denoted $\operatorname{Si}$ , is defined as the [[defining ingredient::def..."