sisI**:ProphaseI(crossingoverandsynapsis),metaphaseI,anaphaseI,telophaseI.**MeiosisII**:Similartomitosisbutwithhaploidcells.**GeneticVariationandC
rossingOver**:Mechanismsthatcontributetogeneticdiversityduringmeiosis,includingcrossingover(exchangeofgeneticmaterialbetweenhomologouschromosomes)andindependentass
ortmentofchromosomes.**FertilizationandGeneticDiversity**:Howfertilizationcombinesgeneticmaterialfromtwoparentstogenerateuniquegeneticcombinationsinoffspring.**Mende
lianGeneticsandMeiosis**:IntegrationofMendelianprinciples(lawsofsegregationandindependentassortment)withmeioticprocessestoexplaininheritancepatterns.**SexChromosomesa
ndSexDetermination**:Explanationofsexdeterminationsystems,includingXX/XY(mammals),ZZ/ZW(birds,somereptiles),andenvironmentalfactors(temperature-
dependentsexdetermination).**GeneticDisordersandChromosomalAbnormalities**:Overviewofgeneticdisorderscausedbyerrorsinmeiosis,suchasnondisjunction(failureofchromoso
mestoseparateproperly),trisomies(e.g.,Downsyndrome),andmonosomies.**HumanReproductionandGametogenesis**:Overviewofhumanreproductiveanatomy,gameteproduction(sp
ermatogenesisandoogenesis),andtherolesofhormones(e.g.,testosterone,estrogen)inreproductivedevelopment.**EmergingTopicsinMeiosisandReproduction**:Introductiontocurrentr
esearchtopicsandadvancementsinmeiosis,reproductivetechnologies(e.g.,invitrofertilization,geneediting),andethicalconsiderations.**SummaryandKeyConcepts**:Thechapterconclu
deswithasummaryofkeyconceptsrelatedtomeiosisandthegeneticbasisofsexualreproduction,reinforcingunderstandingoftheprocessesthatunderpingeneticdiversityandinheritance.Chapt
er9providesacomprehensiveexplorationofmeiosis,geneticvariation,andthegeneticbasisofsexualreproduction.Itbuildsuponfoundationalknowledgeofcelldivision(coveredinChapter8)a
ndpreparesstudentsforfurtherexplorationintogenetics,developmentalbiology,andevolutionaryprocessescoveredinsubsequentchaptersofthetextbook.Ifyouhavespecificquestionsabouta
nyofthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter9,feelfreetoask!Chapter10Chapter10typicallycovers"PatternsofInheritance"inbiologyorgenetics.H
ere'sageneraloverviewofwhatthischapteroftenincludes:**MendelianGenetics**:ItstartswiththebasicprinciplesofinheritanceasdiscoveredbyGregorMendel,suchasdominantandrecessi
vealleles,andthelawsofsegregationandindependentassortment.**ExtensionsofMendelianGenetics**:**IncompleteDominance**:Whereneitheralleleiscompletelydominant.**Codom
inance**:Wherebothallelescontributetothephenotype.**MultipleAlleles**:Genesthathavemorethantwoalleles.**PolygenicInheritance**:Traitsinfluencedbymultiplegenes.**Sex-
linkedInheritance**:Inheritancepatternswheregenesarelocatedonthesexchromosomes(XandYchromosomes).**PedigreeAnalysis**:Howtointerpretfamilytreestodeducepatternsofin
heritanceandpredictprobabilitiesofgenetictraitsinoffspring.**Non-
MendelianInheritance**:IncludesexceptionstoMendel'slawssuchasepistasis(interactionbetweengenes)andenvironmentalinfluencesongeneexpression.**GeneticDisorders**:Exampl
esofgeneticdiseasesanddisorders (interactionenvironmentalinfluencesongeneexpression.**GeneticDisorders**:Examplesofgeneticdiseasesanddisorders
MATICAL
METHODS FOR
PHYSICISTS
A Comprehensive Guide
SEVENTH EDITION
George B. Arfken
Miami University
Oxford, OH
Hans J. Weber
University of Virginia
Charlottesville, VA
Frank E. Harris
University of Utah, Salt Lake City, UT
and
University of Florida, Gainesville, FL
, CHAPTER 1
MATHEMATICAL
PRELIMINARIES
This introductory chapter surveys a number of mathematical techniques that are needed
throughout the book. Some of the topics (e.g., complex variables) are treated in more detail
in later chapters, and the short survey of special functions in this chapter is supplemented
by extensive later discussion of those of particular importance in physics (e.g., Bessel func-
tions). A later chapter on miscellaneous mathematical topics deals with material requiring
more background than is assumed at this point. The reader may note that the Additional
Readings at the end of this chapter include a number of general references on mathemati-
cal methods, some of which are more advanced or comprehensive than the material to be
found in this book.
Overviewofregulatoryframeworksandagenciesresponsibleforoverseeingbiotechnologicalresearch,development,andapplications(e.g.,FDA,USDA).**EmergingTechnologiesinBiotec
hnology**:Introductiontocurrentandemergingtechnologiesinbiotechnologyandgenomics,suchassyntheticbiology,CRISPR-
basedgenomeediting,andbioinformatics.**ImpactofBiotechnologyonSociety**:Examinationofthebroaderimpactsofbiotechnologyonsociety,includingeconomicimplications,jobcreat
ion,andpublicperceptions.**FutureDirectionsinBiotechnology**:Speculationonfuturetrendsandpotentialbreakthroughsinbiotechnology,includingapplicationsinspaceexploration,env
ironmentalsustainability,andbeyond.**SummaryandKeyConcepts**:Thechapterconcludeswithasummaryofkeyconceptsrelatedtobiotechnologyandgenomics,reinforcingunderstandi
ngoftheapplications,implications,andethicalconsiderationsassociatedwithbiotechnologicaladvancements.Chapter12providesacomprehensiveexplorationofthediverseapplicationsand
ethicalconsiderationsofbiotechnology,withafocusongenomicsanditsimpactonvariousaspectsofhumanhealth,agriculture,industry,andsociety.Itbuildsuponfoundationalknowledgeofmo
leculargeneticsandpreparesstudentsforfurtherexplorationintoadvancedtopicsingenetics,biotechnology,andbioethicscoveredinsubsequentchaptersofthetextbook.Ifyouhavespecificque
stionsaboutanyofthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter12,feelfreetoask!Chapter13:MutationsandGeneticTestingChapter13of"EssentialsofBi
ology"bySylviaMaderandMichaelWindelspecht,titled"MutationsandGeneticTesting,"typicallyfocusesonthenatureofmutations,theircauses,consequences,andtheroleofgenetictestingi
nidentifyinggeneticdisorders.Here‘sanoverviewofwhatyoumightfindinthischapter:**IntroductiontoMutations**:DefinitionofmutationsaschangesintheDNAsequencethatcanalterthes
tructureandfunctionofproteinsencodedbygenes.**TypesofMutations**:Explanationofdifferenttypesofmutations:**PointMutations**:Changesinasinglenucleotide(substitutions,inser
tions,deletions).**ChromosomalMutations**:Structuralchangesinchromosomes(e.g.,deletions,duplications,inversions,translocations).**Mutagens**:Environmentalfactors(chemica
ls,radiation)thatcanincreasetherateofmutations.**CausesofMutations**:Overviewofspontaneousmutations(arisingfromerrorsinDNAreplicationorrepair)andinducedmutations(cause
dbymutagens).**ConsequencesofMutations**:Impactofmutationsonproteinstructureandfunction:**SilentMutations**:Nochangeinaminoacidsequence.**MissenseMutations**:Cha
ngeinoneaminoacid.**NonsenseMutations**:Prematureterminationofproteinsynthesis.**FrameshiftMutations**:Insertionordeletionofnucleotides,leadingtoashiftinthereadingframe
.**GeneticTesting**:DefinitionofgenetictestingastheanalysisofDNA,RNA,orchromosomestodetectgeneticvariationsassociatedwithinheriteddisordersorpredispositionstodiseases.**
TypesofGeneticTesting**:Overviewofdifferen
1.1 I NFINITE S ERIES
Perhaps the most widely used technique in the physicist‘s toolbox is the use of infinite
series (i.e., sums consisting formally of an infinite number of terms) to represent functions,
to bring them to forms facilitating further analysis, or even as a prelude to numerical eval-
uation. The acquisition of skill in creating and manipulating series expansions is therefore
an absolutely essential part of the training of one who seeks competence in the mathemat-
ical methods of physics, and it is therefore the first topic in this text. An important part of
this skill set is the ability to recognize the functions represented by commonly encountered
expansions, and it is also of importance to understand issues related to the convergence of
infinite series.
1
Mathematical Methods for Physicists. DOI: 10.1016/B978-0-12-384654-9.00001-3
© 2013 Elsevier Inc. All rights reserved.
,2 Chapler 1 Malhemalical Preliminaries
Fundamental Concepts
The usual way of assigning a meaning to the sum of an infinite number of terms is by
introducing the notion of partial sums. If we have an infinite sequence of terms u1, u2, u3,
u4, u5, . . . , we define the i th partial sum as
i
si = un. (1.1)
n=1
This is a finite summation and offers no difficulties. If the partial sums si converge to a
finite limit as i → ∞,
lim si = S, (1.2)
i
∞ u is said to be convergent and to have the value S. Note that
the infinite series n=1 n
we define the infinite series as equal to S and that a necessary condition for convergence
to a limit is that limn→∞ un =0. This condition, however, is not sufficient to guarantee
convergence.
Sometimes it is convenient to apply the condition in Eq. (1.2) in a form called the
Cauchy criterion, namely that for each ε > 0 there is a fixed number N such that
s j si < ε for all i and j greater than N . This means that the partial sums must cluster
together as we move far out in the sequence.
Some series diverge, meaning that the sequence of partial sums approaches±∞ ; others
may have partial sums that oscillate between two values, as for example,
∞
un = 1 − 1 + 1 − 1 + 1 − · · · − (−1) + · · · .
n=1
This series does not converge to a limit, and can be called oscillatory. Often the term
divergent is extended to include oscillatory series as well. It is important to be able to
determine whether, or under what conditions, a series we would like to use is convergent.
Overviewofregulatoryframeworksandagenciesresponsibleforoverseeingbiotechnologicalresearch,development,andapplications(e.g.,FDA,USDA).**E
mergingTechnologiesinBiotechnology**:Introductiontocurrentandemergingtechnologiesinbiotechnologyandgenomics,suchassyntheticbiology,CRIS
PR-
basedgenomeediting,andbioinformatics.**ImpactofBiotechnologyonSociety**:Examinationofthebroaderimpactsofbiotechnologyonsociety,including
economicimplications,jobcreation,andpublicperceptions.**FutureDirectionsinBiotechnology**:Speculationonfuturetrendsandpotentialbreakthroughs
inbiotechnology,includingapplicationsinspaceexploration,environmentalsustainability,andbeyond.**SummaryandKeyConcepts**:Thechapterconclu
deswithasummaryofkeyconceptsrelatedtobiotechnologyandgenomics,reinforcingunderstandingoftheapplications,implications,andethicalconsideration
sassociatedwithbiotechnologicaladvancements.Chapter12providesacomprehensiveexplorationofthediverseapplicationsandethicalconsiderationsofbiot
echnology,withafocusongenomicsanditsimpactonvariousaspectsofhumanhealth,agriculture,industry,andsociety.Itbuildsuponfoundationalknowledgeof
moleculargeneticsandpreparesstudentsforfurtherexplorationintoadvancedtopicsingenetics,biotechnology,andbioethicscoveredinsubsequentchaptersoft
hetextbook.IfyouhavespecificquestionsaboutanyofthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter12,feelfreetoask!Cha
pter13:MutationsandGeneticTestingChapter13of"EssentialsofBiology"bySylviaMaderandMichaelWindelspecht,titled"MutationsandGenet icTesting,"
typicallyfocusesonthenatureofmutations,theircauses,consequences,andtheroleofgenetictestinginidentifyinggeneticdisorders.Here‘sanoverviewofwhat
youmightfindinthischapter:**IntroductiontoMutations**:DefinitionofmutationsaschangesintheDNAsequencethatcanalterthestructureandfunctionofpr
oteinsencodedbygenes.**TypesofMutations**:Explanationofdifferenttypesofmutations:**PointMutations**:Changesinasinglenucleotide(substitution
s,insertions,deletions).**ChromosomalMutations**:Structuralchangesinchromosomes(e.g.,deletions,duplications,inversions,translocations).**Mutag
ens**:Environmentalfactors(chemicals,radiation)thatcanincreasetherateofmutations.**CausesofMutations**:Overviewofspontaneousmutations(arisi
ngfromerrorsinDNAreplicationorrepair)andinducedmutations(causedbymutagens).**ConsequencesofMutations**:Impactofmutationsonproteinstruct
ureandfunction:**SilentMutations**:Nochangeinaminoacidsequence.**MissenseMutations**:Changeinoneaminoacid.**NonsenseMutations**:Pre
matureterminationofproteinsynthesis.**FrameshiftMutations**:Insertionordeletionofnucleotides,leadingtoashiftinthereadingframe.**GeneticTesting
**:DefinitionofgenetictestingastheanalysisofDNA,RNA,orchromosomestodetectgeneticvariationsassociatedwithinheriteddisordersorpredispositionst
odiseases.**TypesofGeneticTesting**:Overviewofdifferen
Example 1.1.1 THE GEOMETRIC 5ERIE5
The geometric series, starting with u0 = 1 and with a ratio of successive terms r =
un+1/un, has the form
1 + r + r 2 + r 3 + · · · + rn−1 + · · · .
, Its nth partial sum sn (that of the first n terms) is1
1 − rn
sn = . (1.3)
1 −r
Restricting attention to r < 1, so that for large n, r n approaches zero, and sn possesses
the limit
nction:**SilentMutations**:Nochangeinaminoacidsequence.**MissenseM
utations**:Changeinoneaminoacid.**NonsenseMutations**:Prematureter
minationofproteinsynthesis.**FrameshiftMutations**:Insertionordeletiono
fnucleotides,leadingtoashiftinthereadingframe.**GeneticTesting**:Definit
Σ ionofgenetictestingastheanalysisofDNA,RNA,orchromosomestodetectgene
1Multiply and divide sn = n−1 rm by 1 − r . ticvariationsassociatedwithinheriteddisordersorpredispositionstodiseases.*
Overviewofregulatoryframeworksandagenc *TypesofGeneticTesting**:Overviewofdifferen
iesresponsibleforoverseeingbiotechnologic
alresearch,development,andapplications(e.
g.,FDA,USDA).**EmergingTechnologiesi
nBiotechnology**:Introductiontocurrentan
demergingtechnologiesinbiotechnologyand
genomics,suchassyntheticbiology,CRISPR-
basedgenomeediting,andbioinformatics.**I
mpactofBiotechnologyonSociety**:Exami
nationofthebroaderimpactsofbiotechnology
onsociety,includingeconomicimplications,j
obcreation,andpublicperceptions.**Future
DirectionsinBiotechnology**:Speculationo
nfuturetrendsandpotentialbreakthroughsinb
iotechnology,includingapplicationsinspace
exploration,environmentalsustainability,an
dbeyond.**SummaryandKeyConcepts**:T
hechapterconcludeswithasummaryofkeyco
nceptsrelatedtobiotechnologyandgenomics,
reinforcingunderstandingoftheapplications,
implications,andethicalconsiderationsassoc
iatedwithbiotechnologicaladvancements.Ch
apter12providesacomprehensiveexploration
ofthediverseapplicationsandethicalconsider
ationsofbiotechnology,withafocusongenom
icsanditsimpactonvariousaspectsofhumanh
ealth,agriculture,industry,andsociety.Itbuild
suponfoundationalknowledgeofmolecularg
eneticsandpreparesstudentsforfurtherexplor
ationintoadvancedtopicsingenetics,biotechn
ology,andbioethicscoveredinsubsequentcha
ptersofthetextbook.Ifyouhavespecificquesti
onsaboutanyofthesetopicsorwouldlikemore
detailedinformationonaparticularaspectofC
hapter12,feelfreetoask!Chapter13:Mutation
sandGeneticTestingChapter13of"Essentials
ofBiology"bySylviaMaderandMichaelWin
delspecht,titled"MutationsandGeneticTesti
ng,"typicallyfocusesonthenatureofmutation
s,theircauses,consequences,andtheroleofge
netictestinginidentifyinggeneticdisorders.H
ere‘sanoverviewofwhatyoumightfindinthisc
hapter:**IntroductiontoMutations**:Defini
tionofmutationsaschangesintheDNAsequen
cethatcanalterthestructureandfunctionofprot
einsencodedbygenes.**TypesofMutations*
*:Explanationofdifferenttypesofmutations:
**PointMutations**:Changesinasinglenucl
eotide(substitutions,insertions,deletions).**
ChromosomalMutations**:Structuralchang
esinchromosomes(e.g.,deletions,duplicatio
ns,inversions,translocations).**Mutagens*
*:Environmentalfactors(chemicals,radiatio
n)thatcanincreasetherateofmutations.**Cau
sesofMutations**:Overviewofspontaneous
mutations(arisingfromerrorsinDNAreplicati
onorrepair)andinducedmutations(causedby
mutagens).**ConsequencesofMutations**:
Impactofmutationsonproteinstructureandfu
rossingOver**:Mechanismsthatcontributetogeneticdiversityduringmeiosis,includingcrossingover(exchangeofgeneticmaterialbetweenhomologouschromosomes)andindependentass
ortmentofchromosomes.**FertilizationandGeneticDiversity**:Howfertilizationcombinesgeneticmaterialfromtwoparentstogenerateuniquegeneticcombinationsinoffspring.**Mende
lianGeneticsandMeiosis**:IntegrationofMendelianprinciples(lawsofsegregationandindependentassortment)withmeioticprocessestoexplaininheritancepatterns.**SexChromosomesa
ndSexDetermination**:Explanationofsexdeterminationsystems,includingXX/XY(mammals),ZZ/ZW(birds,somereptiles),andenvironmentalfactors(temperature-
dependentsexdetermination).**GeneticDisordersandChromosomalAbnormalities**:Overviewofgeneticdisorderscausedbyerrorsinmeiosis,suchasnondisjunction(failureofchromoso
mestoseparateproperly),trisomies(e.g.,Downsyndrome),andmonosomies.**HumanReproductionandGametogenesis**:Overviewofhumanreproductiveanatomy,gameteproduction(sp
ermatogenesisandoogenesis),andtherolesofhormones(e.g.,testosterone,estrogen)inreproductivedevelopment.**EmergingTopicsinMeiosisandReproduction**:Introductiontocurrentr
esearchtopicsandadvancementsinmeiosis,reproductivetechnologies(e.g.,invitrofertilization,geneediting),andethicalconsiderations.**SummaryandKeyConcepts**:Thechapterconclu
deswithasummaryofkeyconceptsrelatedtomeiosisandthegeneticbasisofsexualreproduction,reinforcingunderstandingoftheprocessesthatunderpingeneticdiversityandinheritance.Chapt
er9providesacomprehensiveexplorationofmeiosis,geneticvariation,andthegeneticbasisofsexualreproduction.Itbuildsuponfoundationalknowledgeofcelldivision(coveredinChapter8)a
ndpreparesstudentsforfurtherexplorationintogenetics,developmentalbiology,andevolutionaryprocessescoveredinsubsequentchaptersofthetextbook.Ifyouhavespecificquestionsabouta
nyofthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter9,feelfreetoask!Chapter10Chapter10typicallycovers"PatternsofInheritance"inbiologyorgenetics.H
ere'sageneraloverviewofwhatthischapteroftenincludes:**MendelianGenetics**:ItstartswiththebasicprinciplesofinheritanceasdiscoveredbyGregorMendel,suchasdominantandrecessi
vealleles,andthelawsofsegregationandindependentassortment.**ExtensionsofMendelianGenetics**:**IncompleteDominance**:Whereneitheralleleiscompletelydominant.**Codom
inance**:Wherebothallelescontributetothephenotype.**MultipleAlleles**:Genesthathavemorethantwoalleles.**PolygenicInheritance**:Traitsinfluencedbymultiplegenes.**Sex-
linkedInheritance**:Inheritancepatternswheregenesarelocatedonthesexchromosomes(XandYchromosomes).**PedigreeAnalysis**:Howtointerpretfamilytreestodeducepatternsofin
heritanceandpredictprobabilitiesofgenetictraitsinoffspring.**Non-
MendelianInheritance**:IncludesexceptionstoMendel'slawssuchasepistasis(interactionbetweengenes)andenvironmentalinfluencesongeneexpression.**GeneticDisorders**:Exampl
esofgeneticdiseasesanddisorders (interactionenvironmentalinfluencesongeneexpression.**GeneticDisorders**:Examplesofgeneticdiseasesanddisorders
MATICAL
METHODS FOR
PHYSICISTS
A Comprehensive Guide
SEVENTH EDITION
George B. Arfken
Miami University
Oxford, OH
Hans J. Weber
University of Virginia
Charlottesville, VA
Frank E. Harris
University of Utah, Salt Lake City, UT
and
University of Florida, Gainesville, FL
, CHAPTER 1
MATHEMATICAL
PRELIMINARIES
This introductory chapter surveys a number of mathematical techniques that are needed
throughout the book. Some of the topics (e.g., complex variables) are treated in more detail
in later chapters, and the short survey of special functions in this chapter is supplemented
by extensive later discussion of those of particular importance in physics (e.g., Bessel func-
tions). A later chapter on miscellaneous mathematical topics deals with material requiring
more background than is assumed at this point. The reader may note that the Additional
Readings at the end of this chapter include a number of general references on mathemati-
cal methods, some of which are more advanced or comprehensive than the material to be
found in this book.
Overviewofregulatoryframeworksandagenciesresponsibleforoverseeingbiotechnologicalresearch,development,andapplications(e.g.,FDA,USDA).**EmergingTechnologiesinBiotec
hnology**:Introductiontocurrentandemergingtechnologiesinbiotechnologyandgenomics,suchassyntheticbiology,CRISPR-
basedgenomeediting,andbioinformatics.**ImpactofBiotechnologyonSociety**:Examinationofthebroaderimpactsofbiotechnologyonsociety,includingeconomicimplications,jobcreat
ion,andpublicperceptions.**FutureDirectionsinBiotechnology**:Speculationonfuturetrendsandpotentialbreakthroughsinbiotechnology,includingapplicationsinspaceexploration,env
ironmentalsustainability,andbeyond.**SummaryandKeyConcepts**:Thechapterconcludeswithasummaryofkeyconceptsrelatedtobiotechnologyandgenomics,reinforcingunderstandi
ngoftheapplications,implications,andethicalconsiderationsassociatedwithbiotechnologicaladvancements.Chapter12providesacomprehensiveexplorationofthediverseapplicationsand
ethicalconsiderationsofbiotechnology,withafocusongenomicsanditsimpactonvariousaspectsofhumanhealth,agriculture,industry,andsociety.Itbuildsuponfoundationalknowledgeofmo
leculargeneticsandpreparesstudentsforfurtherexplorationintoadvancedtopicsingenetics,biotechnology,andbioethicscoveredinsubsequentchaptersofthetextbook.Ifyouhavespecificque
stionsaboutanyofthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter12,feelfreetoask!Chapter13:MutationsandGeneticTestingChapter13of"EssentialsofBi
ology"bySylviaMaderandMichaelWindelspecht,titled"MutationsandGeneticTesting,"typicallyfocusesonthenatureofmutations,theircauses,consequences,andtheroleofgenetictestingi
nidentifyinggeneticdisorders.Here‘sanoverviewofwhatyoumightfindinthischapter:**IntroductiontoMutations**:DefinitionofmutationsaschangesintheDNAsequencethatcanalterthes
tructureandfunctionofproteinsencodedbygenes.**TypesofMutations**:Explanationofdifferenttypesofmutations:**PointMutations**:Changesinasinglenucleotide(substitutions,inser
tions,deletions).**ChromosomalMutations**:Structuralchangesinchromosomes(e.g.,deletions,duplications,inversions,translocations).**Mutagens**:Environmentalfactors(chemica
ls,radiation)thatcanincreasetherateofmutations.**CausesofMutations**:Overviewofspontaneousmutations(arisingfromerrorsinDNAreplicationorrepair)andinducedmutations(cause
dbymutagens).**ConsequencesofMutations**:Impactofmutationsonproteinstructureandfunction:**SilentMutations**:Nochangeinaminoacidsequence.**MissenseMutations**:Cha
ngeinoneaminoacid.**NonsenseMutations**:Prematureterminationofproteinsynthesis.**FrameshiftMutations**:Insertionordeletionofnucleotides,leadingtoashiftinthereadingframe
.**GeneticTesting**:DefinitionofgenetictestingastheanalysisofDNA,RNA,orchromosomestodetectgeneticvariationsassociatedwithinheriteddisordersorpredispositionstodiseases.**
TypesofGeneticTesting**:Overviewofdifferen
1.1 I NFINITE S ERIES
Perhaps the most widely used technique in the physicist‘s toolbox is the use of infinite
series (i.e., sums consisting formally of an infinite number of terms) to represent functions,
to bring them to forms facilitating further analysis, or even as a prelude to numerical eval-
uation. The acquisition of skill in creating and manipulating series expansions is therefore
an absolutely essential part of the training of one who seeks competence in the mathemat-
ical methods of physics, and it is therefore the first topic in this text. An important part of
this skill set is the ability to recognize the functions represented by commonly encountered
expansions, and it is also of importance to understand issues related to the convergence of
infinite series.
1
Mathematical Methods for Physicists. DOI: 10.1016/B978-0-12-384654-9.00001-3
© 2013 Elsevier Inc. All rights reserved.
,2 Chapler 1 Malhemalical Preliminaries
Fundamental Concepts
The usual way of assigning a meaning to the sum of an infinite number of terms is by
introducing the notion of partial sums. If we have an infinite sequence of terms u1, u2, u3,
u4, u5, . . . , we define the i th partial sum as
i
si = un. (1.1)
n=1
This is a finite summation and offers no difficulties. If the partial sums si converge to a
finite limit as i → ∞,
lim si = S, (1.2)
i
∞ u is said to be convergent and to have the value S. Note that
the infinite series n=1 n
we define the infinite series as equal to S and that a necessary condition for convergence
to a limit is that limn→∞ un =0. This condition, however, is not sufficient to guarantee
convergence.
Sometimes it is convenient to apply the condition in Eq. (1.2) in a form called the
Cauchy criterion, namely that for each ε > 0 there is a fixed number N such that
s j si < ε for all i and j greater than N . This means that the partial sums must cluster
together as we move far out in the sequence.
Some series diverge, meaning that the sequence of partial sums approaches±∞ ; others
may have partial sums that oscillate between two values, as for example,
∞
un = 1 − 1 + 1 − 1 + 1 − · · · − (−1) + · · · .
n=1
This series does not converge to a limit, and can be called oscillatory. Often the term
divergent is extended to include oscillatory series as well. It is important to be able to
determine whether, or under what conditions, a series we would like to use is convergent.
Overviewofregulatoryframeworksandagenciesresponsibleforoverseeingbiotechnologicalresearch,development,andapplications(e.g.,FDA,USDA).**E
mergingTechnologiesinBiotechnology**:Introductiontocurrentandemergingtechnologiesinbiotechnologyandgenomics,suchassyntheticbiology,CRIS
PR-
basedgenomeediting,andbioinformatics.**ImpactofBiotechnologyonSociety**:Examinationofthebroaderimpactsofbiotechnologyonsociety,including
economicimplications,jobcreation,andpublicperceptions.**FutureDirectionsinBiotechnology**:Speculationonfuturetrendsandpotentialbreakthroughs
inbiotechnology,includingapplicationsinspaceexploration,environmentalsustainability,andbeyond.**SummaryandKeyConcepts**:Thechapterconclu
deswithasummaryofkeyconceptsrelatedtobiotechnologyandgenomics,reinforcingunderstandingoftheapplications,implications,andethicalconsideration
sassociatedwithbiotechnologicaladvancements.Chapter12providesacomprehensiveexplorationofthediverseapplicationsandethicalconsiderationsofbiot
echnology,withafocusongenomicsanditsimpactonvariousaspectsofhumanhealth,agriculture,industry,andsociety.Itbuildsuponfoundationalknowledgeof
moleculargeneticsandpreparesstudentsforfurtherexplorationintoadvancedtopicsingenetics,biotechnology,andbioethicscoveredinsubsequentchaptersoft
hetextbook.IfyouhavespecificquestionsaboutanyofthesetopicsorwouldlikemoredetailedinformationonaparticularaspectofChapter12,feelfreetoask!Cha
pter13:MutationsandGeneticTestingChapter13of"EssentialsofBiology"bySylviaMaderandMichaelWindelspecht,titled"MutationsandGenet icTesting,"
typicallyfocusesonthenatureofmutations,theircauses,consequences,andtheroleofgenetictestinginidentifyinggeneticdisorders.Here‘sanoverviewofwhat
youmightfindinthischapter:**IntroductiontoMutations**:DefinitionofmutationsaschangesintheDNAsequencethatcanalterthestructureandfunctionofpr
oteinsencodedbygenes.**TypesofMutations**:Explanationofdifferenttypesofmutations:**PointMutations**:Changesinasinglenucleotide(substitution
s,insertions,deletions).**ChromosomalMutations**:Structuralchangesinchromosomes(e.g.,deletions,duplications,inversions,translocations).**Mutag
ens**:Environmentalfactors(chemicals,radiation)thatcanincreasetherateofmutations.**CausesofMutations**:Overviewofspontaneousmutations(arisi
ngfromerrorsinDNAreplicationorrepair)andinducedmutations(causedbymutagens).**ConsequencesofMutations**:Impactofmutationsonproteinstruct
ureandfunction:**SilentMutations**:Nochangeinaminoacidsequence.**MissenseMutations**:Changeinoneaminoacid.**NonsenseMutations**:Pre
matureterminationofproteinsynthesis.**FrameshiftMutations**:Insertionordeletionofnucleotides,leadingtoashiftinthereadingframe.**GeneticTesting
**:DefinitionofgenetictestingastheanalysisofDNA,RNA,orchromosomestodetectgeneticvariationsassociatedwithinheriteddisordersorpredispositionst
odiseases.**TypesofGeneticTesting**:Overviewofdifferen
Example 1.1.1 THE GEOMETRIC 5ERIE5
The geometric series, starting with u0 = 1 and with a ratio of successive terms r =
un+1/un, has the form
1 + r + r 2 + r 3 + · · · + rn−1 + · · · .
, Its nth partial sum sn (that of the first n terms) is1
1 − rn
sn = . (1.3)
1 −r
Restricting attention to r < 1, so that for large n, r n approaches zero, and sn possesses
the limit
nction:**SilentMutations**:Nochangeinaminoacidsequence.**MissenseM
utations**:Changeinoneaminoacid.**NonsenseMutations**:Prematureter
minationofproteinsynthesis.**FrameshiftMutations**:Insertionordeletiono
fnucleotides,leadingtoashiftinthereadingframe.**GeneticTesting**:Definit
Σ ionofgenetictestingastheanalysisofDNA,RNA,orchromosomestodetectgene
1Multiply and divide sn = n−1 rm by 1 − r . ticvariationsassociatedwithinheriteddisordersorpredispositionstodiseases.*
Overviewofregulatoryframeworksandagenc *TypesofGeneticTesting**:Overviewofdifferen
iesresponsibleforoverseeingbiotechnologic
alresearch,development,andapplications(e.
g.,FDA,USDA).**EmergingTechnologiesi
nBiotechnology**:Introductiontocurrentan
demergingtechnologiesinbiotechnologyand
genomics,suchassyntheticbiology,CRISPR-
basedgenomeediting,andbioinformatics.**I
mpactofBiotechnologyonSociety**:Exami
nationofthebroaderimpactsofbiotechnology
onsociety,includingeconomicimplications,j
obcreation,andpublicperceptions.**Future
DirectionsinBiotechnology**:Speculationo
nfuturetrendsandpotentialbreakthroughsinb
iotechnology,includingapplicationsinspace
exploration,environmentalsustainability,an
dbeyond.**SummaryandKeyConcepts**:T
hechapterconcludeswithasummaryofkeyco
nceptsrelatedtobiotechnologyandgenomics,
reinforcingunderstandingoftheapplications,
implications,andethicalconsiderationsassoc
iatedwithbiotechnologicaladvancements.Ch
apter12providesacomprehensiveexploration
ofthediverseapplicationsandethicalconsider
ationsofbiotechnology,withafocusongenom
icsanditsimpactonvariousaspectsofhumanh
ealth,agriculture,industry,andsociety.Itbuild
suponfoundationalknowledgeofmolecularg
eneticsandpreparesstudentsforfurtherexplor
ationintoadvancedtopicsingenetics,biotechn
ology,andbioethicscoveredinsubsequentcha
ptersofthetextbook.Ifyouhavespecificquesti
onsaboutanyofthesetopicsorwouldlikemore
detailedinformationonaparticularaspectofC
hapter12,feelfreetoask!Chapter13:Mutation
sandGeneticTestingChapter13of"Essentials
ofBiology"bySylviaMaderandMichaelWin
delspecht,titled"MutationsandGeneticTesti
ng,"typicallyfocusesonthenatureofmutation
s,theircauses,consequences,andtheroleofge
netictestinginidentifyinggeneticdisorders.H
ere‘sanoverviewofwhatyoumightfindinthisc
hapter:**IntroductiontoMutations**:Defini
tionofmutationsaschangesintheDNAsequen
cethatcanalterthestructureandfunctionofprot
einsencodedbygenes.**TypesofMutations*
*:Explanationofdifferenttypesofmutations:
**PointMutations**:Changesinasinglenucl
eotide(substitutions,insertions,deletions).**
ChromosomalMutations**:Structuralchang
esinchromosomes(e.g.,deletions,duplicatio
ns,inversions,translocations).**Mutagens*
*:Environmentalfactors(chemicals,radiatio
n)thatcanincreasetherateofmutations.**Cau
sesofMutations**:Overviewofspontaneous
mutations(arisingfromerrorsinDNAreplicati
onorrepair)andinducedmutations(causedby
mutagens).**ConsequencesofMutations**:
Impactofmutationsonproteinstructureandfu
