`MATHEMATICAL FUNCTIONS
`
`WITH FORMULAS, GRAPHS,
`AND MATHEMATICAL TABLES
`
`·}
`
`Milton Abramowitz and Irene A. Stegun
`
`Edited by
`
`DOVER PUBLICATIONS, INC., NEW YORK
`
`Petitioner Sirius XM Radio Inc. - Ex. 1024, p. 1
`
`
`
`The text relating to physical constants and conversion factors (page 6)
`has been modified to take into account the newly adopted Systeme Interna(cid:173)
`tional d'Unites (SI).
`
`ERRATA NOTICE
`
`The original printing of this Handbook (June 1964) contained
`errors that have been corrected in the reprinted editions. These cor(cid:173)
`rections are marked with an asterisk (*) for identification. The errors
`occurred on the following pages: 2-3, 6-8, 10, 15, 19-20,25, 76, 85, 91, 102,
`187,189-197,218,223,225,233,250,255,260-263,268,271-273,292,302,
`328,332,333-337,362,365,415,423,438-440,443,445,447,449,451,484,
`498,505-506,509-510,543,556,558,562,571,595,599,600,722-723,739,
`742, 744,746,752,756,760-765,774,777-785,790,79~801,822-823,832,
`835,844,886-889,897,914,915,920, 930-931~936,940-941, 944-950,953,
`960,963,989-990,1010,1026.
`
`,.
`
`Published in Canada by General Publishing Com(cid:173)
`pany, Ltd., 30 Lesmill Road , Don Mills, Toronto,
`Ontario.
`Published in the United Kingdom by Constable
`and Company, Ltd., 10 Orange Street, London WC 2.
`
`This Dover edition , first published in 1965, is an
`unabridged and unaltered republication of the work
`originally published by the National Burea u of
`Standards in 1964.
`This fifth Dover printing conforms to the seventh
`(May 1968) printing by the Government Printing
`Office, except that additional corrections have been
`made on pages 18, 79, 80, 82, 408, 450, 722, 786, 825
`and 934.
`
`Standard Book Numb e1·: 486-61272-4
`Library of Congress Catalog Card Number: 65-12253
`
`Manufactured in the United States of America
`Dover Publications, Inc.
`180 Varick Street
`New York, N.Y. 10014
`
`Petitioner Sirius XM Radio Inc. - Ex. 1024, p. 2
`
`
`
`16
`
`ELEMENTARY ANALYTICAL METHODS,
`
`Reversion of Series
`
`3.6.25 Given
`y = a.x+ br+ cx3 +dx'+ ex5 + jx6+ gx7 +
`
`then
`x=Ay+By2+Cy3+Dy4+Ey5+Fy6+Gy7+ ..
`where
`aA=l
`a3B=-b
`a5C=2b 2-ac
`a7D=5abc-a2d-5b3
`a9E=6a2bd+3a2c2+ 14b4-a3e-21ab2c
`a11F=7a3be+7a3cd+84ab3c-a4j
`-28a2bc2-42b5-28a2b2d
`a13G=8a4bf+8a4ce+4a4d2+ 120aWd
`+ 180a2b2c2+ 132b6-a5g-36a3b2e
`-72a3bcd -12a3c3 - 330ab4c
`
`Kummer's Transformation of Series
`
`""
`3.6.26 LetL:ak=s be a given convergent series and
`k=O
`"" L: ck=c be a given convergent series with known
`
`k=O
`sum c such that lim ak=X ;eo.
`k-4"" ck
`
`Then
`
`Polar Form
`
`z=rei 8=r(cos O+i sin 0)
`
`3.7.2
`
`3.7.3
`
`3.7.4 Argument: arg z=arctan (yjx)=O (other
`notations for arg z are am z and ph z).
`
`Real Part: x=£3iz=r cos 0
`
`Imaginary Part: y=Yz=r sin 0
`
`Con1plex Conjugate of 111
`
`lzl=lzl
`-
`arg z=-arg z
`
`Multiplication and Division
`
`arg (z1z2)=arg z1+arg z2
`
`Z1Z2
`Z1
`z2=lz2l 2
`
`xix2+YIY2+i(x2yi....i·xly2)
`~+m
`
`3.7.5
`
`3.7.6
`
`3.7.7
`
`3.7.8
`
`3.7.9
`
`3.7.10
`
`3.7.ll
`
`3.7.12
`
`3.7.13
`
`3.7.14
`
`Euler's Transformation of Series
`
`""
`3.6.27 If L: (-l)ka"=a0-a1 +~- ... is a con-
`k-o
`vergent series with sum s then
`
`3,7,15
`
`arg c::)=arg Z1-arg Z2
`
`Powers
`
`3.7.17
`
`3.7.18
`
`3.7.19
`
`=r" cos nO+ir" sin nO
`(n=O,± 1,±2, ... )
`z2=r-y2+i(2xy)
`
`il=il- 3xif+i(3x2y-y3)
`
`3.7.20
`
`z4=x4 -6rif+y4+i(4ily-4xy3)
`
`3.7.21 z5=x"-10rif+5xy'+i(5:ty-10x2if+y5)
`
`3.7.22
`
`Euler-Maclaurin Summation Formula
`
`3.6.28
`1
`25J"= f" j(k)dk--2
`Jo
`k~l
`1
`(f"'(n)-f"'(0)]+-1-
`[j<Y>(n)-j<v>(o)]
`- -
`720
`30240
`-120~600 (j<VII>(n)-j<YW(Q))+ ...
`
`12
`
`1
`
`(j(O)+J(n)J+
`
`(j'(n)-j'(O))
`
`
`
`3.7. Complex NUJ;n.bers and Functions
`
`3.7.1
`
`Cartesian Forn1
`
`z=x+iy
`
`+i r(~) xn-ly-(;) xn-3if+ ... ],
`
`(n=1,2, ... )
`
`Petitioner Sirius XM Radio Inc. - Ex. 1024, p. 3
`
`
Accessing this document will incur an additional charge of $.
After purchase, you can access this document again without charge.
Accept $ Charge
Still Working On It
This document is taking longer than usual to download. This can happen if we need to contact the court directly to obtain the document and their servers are running slowly.
Give it another minute or two to complete, and then try the refresh button.
A few More Minutes ... Still Working
It can take up to 5 minutes for us to download a document if the court servers are running slowly.
Thank you for your continued patience.
This document could not be displayed.
We could not find this document within its docket. Please go back to the docket page and check the link. If that does not work, go back to the docket and refresh it to pull the newest information.
Your account does not support viewing this document.
You need a Paid Account to view this document. Click here to change your account type.
Your account does not support viewing this document.
Set your membership
status to view this document.
With a Docket Alarm membership, you'll
get a whole lot more, including:
- Up-to-date information for this case.
- Email alerts whenever there is an update.
- Full text search for other cases.
- Get email alerts whenever a new case matches your search.
One Moment Please
The filing “” is large (MB) and is being downloaded.
Please refresh this page in a few minutes to see if the filing has been downloaded. The filing will also be emailed to you when the download completes.
Your document is on its way!
If you do not receive the document in five minutes, contact support at support@docketalarm.com.
Sealed Document
We are unable to display this document, it may be under a court ordered seal.
If you have proper credentials to access the file, you may proceed directly to the court's system using your government issued username and password.
Access Government Site
