Integrals Related to the Error Function

249 Pages·2020·5.014 MB·\249

by  Nikolai E. Korotkov (Author), Alexander N. Korotkov (Author)

INTEGRALS RELATED TO THE
ERROR FUNCTION
Nikolai E. Korotkov and Alexander N. Korotkov
A Chapman & Hall Book
Integrals Related
to the Error Function

Integrals Related
to the Error Function
Nikolai E. Korotkov
Retired Leading Researcher
Voronezh Institute of Communications, Russia
Alexander N. Korotkov
Professor, University of California
Riverside, USA
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2020 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
International Standard Book Number-13: 978-0-367-40820-6 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors
and publishers have attempted to trace the copyright holders of all material reproduced in this
publication and apologize to copyright holders if permission to publish in this form has not been
obtained. If any copyright material has not been acknowledged please write and let us know so we
may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information
storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.
copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC),
222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that
provides licenses and registration for a variety of users. For organizations that have been granted a
photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
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and the CRC Press Web site at
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Contents��
Preface, ix
Acknowledgments, xi
Authors, xiii
Notations and Definitions, xv
CRC Press
Taylor & Francis Group Introduction, xvii
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
Part 1 ◾ Indefinite Integrals 1
© 2020 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business 1.1 INTEGRALS OF THE FORM ∫z nexp ∓ (α z +β)2  dz 1
 
No claim to original U.S. Government works
Printed on acid-free paper 1.2 INTEGRALS OF THE FORM ∫z n exp( ∓ α 2z 2+β z + γ)dz 3
International Standard Book Number-13: 978-0-367-40820-6 (Hardback)
1.3 INTEGRALS OF THE FORM
This book contains information obtained from authentic and highly regarded sources. Reasonable ∫erfn (α z +β)exp −(α z +β)2 dz 5
efforts have been made to publish reliable data and information, but the author and publisher cannot  
assume responsibility for the validity of all materials or the consequences of their use. The authors
and publishers have attempted to trace the copyright holders of all material reproduced in this
1.4 INTEGRALS OF THE FORM
publication and apologize to copyright holders if permission to publish in this form has not been
obtained. If any copyright material has not been acknowledged please write and let us know so we ∫zn erf (α z +β)exp −−(α z +β)2 dz 5
may rectify in any future reprint.  
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, 1.5 INTEGRALS OF THE FORM ∫zn erf (αz+β)exp(βz+γ)dz 7
transmitted,or utilized in any form by any electronic, mechanical, or other means, now known or 1
hereafter invented, including photocopying, microfilming, and recording, or in any information
storageorretrievalsystem,without written permission from the publishers. 1.6 INTEGRALS OFTHE FORM
For permission to photocopy or use material electronically from this work, please access www. ∫z 2m+1 erf (α z +β)exp( ∓ α 2z 2+ γ)dz 8
copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC),
222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that
provides licenses and registration for a variety of users. For organizations that have been granted a 1.7 INTEGRALS OF THE FORM
photocopy license by the CCC, a separate system of payment has been arranged. ∫z2m+1erf (α z +β)exp(αz2 + γ)dz 9
1
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation without intent to infringe.
1.8 INTEGRALS OF THE FORM ∫ zn erf2 (α z) exp(∓α 2z2 )dz 11
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
v
http://www.crcpress.com
vi ◾ Contents
1.9 INTEGRALS OF THE FORM ∫znerf(αz+β)dz 12
1.10 INTEGRALS OF THE FORM ∫znerf2(αz+β)dz 13
1.11 INTEGRALS OF THE FORM ∫z2merf(αz)erf(αz)dz 15
1
1.12 INTEGRALS OF THE FORM ∫z2m+1erf3(αz)dz 16

1.13 INTEGRALS OF THE FORMS 
∫znsinm(α2z2+βz+γ)dz,
∫znsinhm(α2z2+βz+γ)dz,

∫zncosm(α2z2+βz+γ)dz,
∫zncoshm(α2z2+βz+γ)dz 17
1.14 INTEGRALS OF THE FORM
∫znsin(α2z2+βz+γ)exp(βz)dz 27
1
1.15 INTEGRALS OF THE FORM
∫znexp(−α2z2+βz)sin(βz+γ)dz 30
1
1.16 INTEGRALS OF THE FORM
∫znexp(−αz2+βz)sin(αz2+βz+γ)dz 34
1 1
1.17 INTEGRALS OF THE FORM
∫znerf(αz+β)exp(βz)sin(β z+γ)dz 38
1 2
1.18 INTEGRALS OF THE FORM
∫z2n+1erf(αz+β)exp(αz2)sin(α z2+γ)dz 44
1 2
Part 2 ◾ Definite Integrals 55
2.1 INTEGRALS OF znexp∓(αz+β)2 55
 
2.2 INTEGRALS OF znexp(∓α2z2 +βz+γ) 58
2.3 INTEGRALS OF erfn(αz+β)exp−(αz+β)2 62
 
Contents ◾ vii
2.4 INTEGRALS OF znerf(αz+β)exp−(αz+β)2 63
 
2.5 INTEGRALS OF znerf(αz+β)exp(βz+γ) 66
1
2.6 INTEGRALS OF z2m+1erf(αz+β)exp(−αz2) 69
1
2.7 INTEGRALS OF znexp(−αz2+βz)erf(αz+β ),
1 1
znexp(−αz2)erf(αz+β )erf(α z+β ) 72
1 1 2 2
2.8 INTEGRALS OF sin2m+1(α2z2+βz+γ),
sinh2m+1(α2z2+βz+γ), cos2m+1(α2z2+βz+γ),

cosh2m+1(α2z2+βz+γ) 87
2.9 INTEGRALS OF znsin(α2z2+βz+γ)exp(βz) 93
1
2.10 INTEGRALS OF
znexp(−α2z2+βz)sin(αz2+βz+γ) 97
1 1
2.11 INTEGRALS OF znerf(αz+β)exp(βz)sin(β z+γ) 105
1 2
2.12 INTEGRALS OF
z2n+1erf(αz+β)exp(−αz2)sin(α z2+γ) 110
1 2
2.13 INTEGRALS OF
znerf(αz+β)exp(−αz2+βz)sin(α z2+β z+γ),
1 1 2 2
znerf(αz)exp(−αz2)sin(βz),
1

znerf(αz)exp(−αz2)cos(βz) 120
1
2.14 INTEGRALS OF zn±1−erf(αz+β),
 
znerf(αz+β )∓erf(α z+β ) 138
 1 1 2 2 
2.15 INTEGRALS OF zn±1−erf(αz+β)2,
 
zn1−erf2(αz+β),
 
znerf(α1z+β1)∓erf2(α2z+β2),
znerf2(αz+β )−erf2(α z+β ) 145
 1 1 2 2 
viii ◾ Contents
2.16 INTEGRALS OF znexp(βz)±1−erf(αz+β ),
 1 
znexp(βz)erf(α1z+β1)∓erf(α2z+β2) 154
2.17 INTEGRALS OF
znexp(−α2z2+βz) ±1−erf(αz+β ),
 1 1 
znexp(−α2z2+βz) erf(α1z+β1)∓erf(α2z+β2) 159
2.18 INTEGRALS OF znerf(αz+β )exp−(α z+β )2,
1  2 2 
zn±1−erf(αz+β )exp−(αz+β )2,
 1   2 
znexp(βz)erf2(αz+β ),
1
znexp(βz)1−erf2(αz+β ),
 1 
znexp(βz)erf2(αz+β )−erf2(α z+β ) 178
 1 1 2 2 
2.19 INTEGRALS OF
(±1)n −erfn(αz+β)exp−(αz+β)2,
   
z2m+1±1−erf(αz)3, z2m+1±1−erf3(αz),
   
z2m+1erf3(αz)∓erf3(α z),
 1 2 
z2merf2(αz)exp(−α2z2),

z2m1−erf2(αz)exp(−α2z2) 200
 
2.20 INTEGRALS OF znsin(βz+γ)±1−erf(αz+β ),
 1 
znsin(βz+γ)erf(α1z+β1)∓erf(α2z+β2),

znsin(αz2+βz+γ)±1−erf(α1z+β1),
znsin(αz2+βz+γ)erf(α1z+β1)∓erf(α2z+β2) 205
APPENDIX 221
Preface
This book presents a table of integrals related to the error function,
including indefinite and improper definite integrals. Since many tables
of integrals have been published previously and, moreover, computers are
widely used nowadays to find integrals numerically and analytically, a nat-
ural question is why such a new table would be useful. There are at least
three reasons for that. First, to the best of our knowledge, this is the first
book (except Russian versions of essentially the same book), which presents
a comprehensive collection of integrals related to the error function. Most
of the formulas in this book have not been presented in other tables of inte-
grals or have been presented only for some special cases of parameters or
for integration only along the real axis of the complex plane. Second, many
of the integrals presented here cannot be obtained using a computer (except
via an approximate numerical integration). Third, for improper integrals,
this book emphasizes the necessary and sufficient conditions for the valid-
ity of the presented formulas, including the trajectory for going to infinity
on the complex plane; such conditions are usually not given in computer-
assisted analytical integration and often not presented in the previously
published tables of integrals.
We hope that this book will be useful to researchers whose work
involves the error function (e.g., via probability integrals in communica-
tion theory). It can also be useful to a broader audience.
Nikolai E. Korotkov
Alexander N. Korotkov
ix