Engineering Mathematics: gamma

Showing posts with label gamma. Show all posts

Showing posts with label gamma. Show all posts

Saturday, June 2, 2012

Dirichlet Beta - Gamma

DBG.ID.1: Integral representation

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given

Dirchelet Beta :

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(1)

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(2)

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(3)

replacing (2) with (3)

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(4)

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(5)

using geometric series

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(6)

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(7)

DBG.ID.2: Integral representation

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proof

multiply both sides by s

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(8)

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(9)

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(10)

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(11)

DBG.ID.3:Recurrence Relation

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given

from DBG.ID.1

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(1)

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(2)

from DBG.ID.2

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(3)

(3) - (2) results in :

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(4)

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(5)

DBG.ID.5: Integral representation by natural log

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from DBG.ID.1

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(1)

Change of variable

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Therefore,

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(2)

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(3)

if s=2

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(4)

Friday, June 1, 2012

Gamma-Polylogarithm

GP.ID.1:

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Proof

Given

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(1)

therefore ,

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(2)

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(3)

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(4)

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(5)

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(6)

Based on the identity:

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(7)

From (5) and (6) ,

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Friday, October 28, 2011

Gamma-Catalan Relation

GC.ID.1: Catalan Number Identity

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Proof

based on recurrence relation of gamma

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(1)

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(2)

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(3)

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(4)

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(5)

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

(6)

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

(7)

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

(8)

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

(9)

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

(10)

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

(11)

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(12)

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(13)

based on the Catalan number identity , therefore ,

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(14)

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(15)

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(16)

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