Appendix A

Sturm-Liouville, Legendre and Bessel Equations

A.1 Sturm-Liouville Equations

A.1.1 Regular equations

The eigenfunctions of a large class of ODEs form complete orthonormal systems in suit-able Hilbert spaces. To this class belong equations of the form

� �p .x/ u0�0 C q .x/ u D �w .x/ u .x/ ; a < x < b (A.1)

under the assumption that p; p0; q; and w are continuous and positive on Œa; b�. In sucha case (A.1) is called a regular Sturm-Liouville equation. We associate to (A.1) theboundary conditions:

˛u .a/ � ˇp .a/ u0 .a/ D 0

u .b/ � ıp .b/ u0 .b/ D 0;(A.2)

where the coefficients ˛; ˇ; ; ı are real numbers. To avoid trivial situations we shall as-sume the following normalization condition:

˛2 C ˇ2 D 2 C ı2 D 1:

In general, problem (A.1), (A.2) has nontrivial solutions only for special values of the pa-rameter �, called eigenvalues. The corresponding solutions are called eigenfunctions, andthey form the eigenspace associated to �. Let us introduce the (Hilbert) space L2w .a; b/of weighted square-integrable functions u on .a; b/with respect to the weight functionw:

L2w .a; b/ D´u W

Z b

a

u2 .x/w .x/ dx < 1μ

.

Then the following theorem holds:

© Springer International Publishing Switzerland 2015S. Salsa, G. Verzini, Partial Differential Equations in Action. Complements and Exercises,UNITEXT – La Matematica per il 3+2 87, DOI 10.1007/978-3-319-15416-9_A

406 Appendix A Sturm-Liouville, Legendre and Bessel Equations

Theorem A.1. There exists an increasing sequence of positive numbers ¹�j ºj�1 such that�j ! C1 and:

a) Problem (A.1), (A.2) admits a non-trivial solution if and only if � equals one of the �j .

b) For every j , �j is simple, that is the associated eigenspace has dimension 1.

c) The eigenfunction system ¹'j ºj�1 (suitably normalised) is an orthonormal basis inL2w .a; b/.

A.1.2 Legendre’s equation

When the coefficient p is zero, e.g. at a or b, the equation is irregular and the studybecomes more complicated. A classical case is that of Legendre’s equation��

1 � x2�u0�0 C �u D 0 � 1 < x < 1: (A.3)

In the applications this is coupled with boundary conditions of the type

u .�1/ finite, u .1/ finite. (A.4)

Particular solutions to (A.3), (A.4) are Legendre’s polynomials, defined by the Rodriguesformula:

Ln .x/ D 1

2nnŠ

dn

dxn

�x2 � 1�n .n � 0/:

Each polynomial Ln corresponds to the eigenvalue �n

D n .nC 1/. The first four Legen-dre polynomials are

L0 .x/ D 1, L1 .x/ D x, L2 .x/ D 1

2

�3x2 � 1� , L4 .x/ D 1

2

�5x3 � 3x� :

The following theorem holds:

Theorem A.2. In relationship to problem (A.3), (A.4):

a) There exists a non-zero solution if and only if

�D�n D n .nC 1/ ; n D 0; 1; 2; : : : :

b) For every n � 0 the solution corrisponding to �n is unique up to a constant factor, andcoincides with Legendre’s polynomial Ln.

c) The normalised polynomials ´r2nC 1

2Ln

μn�0

form an orthonormal system in L2 .�1; 1/.

A.2 Bessel’s Equation and Functions 407

Theorem A.2 allows to expand any f 2 L2 .�1; 1/ in Fourier-Legendre series:

f .x/ D1XnD0

fnLn .x/ ; where fn D 2nC 1

2

Z 1

�1f .x/Ln .x/ ; dx

with L2 .�1; 1/-convergence. We also have a result about pointwise convergence, in per-fect analogy with Fourier series.

Theorem A.3. If f and f 0 have at most a finite number of jump points in .�1; 1/, then

1XnD0

fnLn .x/ D f .xC/C f .x�/2

for every x 2 .�1; 1/.

A.2 Bessel’s Equation and Functions

A.2.1 Bessel functions

Here is a short summary of the main properties of Bessel functions. First, though, we needto introduce a function interpolating the values of the factorial nŠ: The gamma function� D � .z/ is

� .z/ DZ 1

0

e�t tz�1dt (A.5)

for z complex with Re z > 0. The function � is analytic for Re z > 0 and satisfies thefollowing relationships:

� .z C 1/ D z� .z/

� .z/ � .1 � z/ D �

sin�z.z ¤ 0; 1; 2; : : :/:

In particular,� .nC 1/ D nŠ .n D 0; 1; 2; : : :/

and

�

�nC 1

2

D 1 � 3 � 5 � � � � � 2n � 1

2n

p� .n D 1; 2; : : :/ :

One can define � .z/ for z real, negative and not integer, using

� .z/ D � .z C 1/

z:

In fact, we know how to compute � on .0; 1/, and the formula allows to find � on .�1; 0/.In general, once we know� on .�n;�nC 1/, we can compute it on .�n � 1;�n/. Finally,

408 Appendix A Sturm-Liouville, Legendre and Bessel Equations

�4 �3 �2 �1 1 2 3 4

�5

5

Fig. A.1 Graph of the gamma function on the real axis

coherently with (A.5) we define

� .�2n/ D �1 and � .�2n � 1/ D C1:

In this way � is defined on the entire real axis (Fig. A.1).

Bessel’s function of the first kind and order p, p real, is

Jp .z/ D1XkD0

.�1/k� .k C 1/ � .k C p C 1/

�z2

�pC2k.

In particular, if p D n � 0 is an integer (Fig. A.2):

Jn .z/ D1XkD0

.�1/kkŠ .k C n/Š

�z2

�nC2k:

When p D �n is a negative integer, the first n terms of the series vanish and

J�n .z/ D .�1/n Jn .z/ :

Hence Jn .z/ and J�n .z/ are linearly dependent.If p is not integer, for z ! 0 we have asymptotic behaviours:

Jp .z/ D 1

� .1C p/

�z2

�pCO �zpC2� , J�p .z/ D 1

� .1 � p/�z2

��pCO �z�pC2�so Jp .z/ and J�p .z/ are linearly independent.

A.2 Bessel’s Equation and Functions 409

5 10 15 20

�0:5

0:5

1

Fig. A.2 Graphs of J0 (solid), J1 (dashed) and J2 (dotted)

Functions of the first kind satisfy a number of identities:

d

dz

�zpJp .z/

� D zpJp�1 .z/ ,d

dz

�z�pJp .z/

� D �z�pJpC1 .z/ : (A.6)

In particularJ 00 .z/ D �J1 .z/ :

From these we also infer that for p D n C 12

(and only in that case), the correspondingBessel functions are elementary. For instance,

J 12.z/ D

r2

�zsin z, J� 1

2.z/ D

r2

�zcos z.

Particularly important are the zeroes of Jp . For any p, there is an infinite increasing se-quence

®˛pj j̄�1 of positive numbers such that

Jp�˛pj

� D 0 .j D 1; 2; : : :/:

When p is not an integer, every linear combination

c2Jp .z/C c2J�p .z/

is a Bessel function of the second kind. The (standard) function of the second kind is

Yp .z/ D cosp�Jp .z/ � J�p .z/sinp�

.

When p D n is integer, one defines1 (Fig. A.3)

Yn .z/ WD limp!nYp .z/

Note that Yp .z/ ! �1 when z ! 0C.

1 One can prove that the limit exists.

410 Appendix A Sturm-Liouville, Legendre and Bessel Equations

5 10 15 20

�1

�0:5

0:5

Fig. A.3 Graphs of Y0 (solid), Y1 (dashed) and Y2 (dotted)

A.2.2 Bessel’s equation

The Bessel functions Jp ,Yp are solutions of the so-called Bessel equation of order p � 0:

z2y00 C zy0 C �z2 � p2�y D 0:

The general integral is given, for any p � 0, by

y .z/ D c1Jp .z/C c2Yp .z/ .

In the most important applications, one is typically led to solve the (parametric) equation(with parameter �)

z2y00 C zy0 C��2z2 � p2

�y D 0 (A.7)

on a bounded interval .0; a/, with boundary conditions of the sort

y .0/ finite, y .a/ D 0. (A.8)

For these, the following theorem holds.

Theorem A.4. Problem (A.7), (A.8) has nontrivial solutions if and only if

� D �pj D�˛pja

�2.

In that case the solutions are

ypj .z/ D Jp

�˛pjaz�

up to multiplicative constants. Moreover, the normalised functionsp2

aJpC1�˛pj

�ypj

A.2 Bessel’s Equation and Functions 411

form an orthonormal basis in (w.z/ D z)

L2w .0; a/ D²u W kuk22;w D

Z a

0

u2 .z/ zdz < 1³

by virtue of the orthogonality relations:

2

a2J 2pC1�˛pj

� Z a

0

zJp��pj z

�Jp��pkz

�dz D

´0 j ¤ k

1 j D k:

With Theorem A.4 we can expand any f 2 L2w .0; a/ in Fourier-Bessel series:

f .z/ D1XjD1

fjJp��pj z

�; where fj D 2

a2J 2pC1�˛pj

� Z a

0

zf .z/ Jp��pj z

�dz;

with L2w .0; a/-convergence.Let us compute, for example, the expansion of f .x/ D 1 on the interval .0; 1/, with

p D 0:

fj D 2

J 21�˛pj

� Z 1

0

zJ0�˛0j z

�dz:

Using (A.6),d

dzŒzJ1 .z/� D zJ0 .z/

so we may write Z 1

0

zJ0�˛0j z

�dz D

1

�0jzJ1

�˛0j z

��10

D J1�˛0j

��0j

:

Finally

1 D1XjD1

2

�0jJ1�˛0j

�J0 �˛0j z�with convergence in norm L2w .0; 1/.

Also in this case one can insure pointwise convergence.

Theorem A.5. If f and f 0 have at most finitely many jump discontinuities on .0; a/, then

1XjD1

fjJp��pj z

� D f .zC/C f .z�/2

at every point z 2 .0; a/.

Appendix B

Identities

Here is a compilation of significant formulas and identities of common use.

B.1 Gradient, Divergence, Curl, Laplacian

Let F;u; v be vector fields and f; ' scalar fields, all assumed regular on R3.

Orthogonal Cartesian coordinates

1. gradient:

rf D @f

@xi C @f

@yj C @f

@zk

2. divergence:

div F D @

@xFx C @

@yFy C @

@zFz

3. Laplacian:

�f D @2f

@x2C @2f

@y2C @2f

@z2

4. curl:

curl F Dˇ̌̌̌ˇ̌ i j k@x @y @zFx Fy Fz

ˇ̌̌̌ˇ̌

Cylindrical coordinates

x D r cos �; y D r sin �; z D z .r > 0; 0 � � � 2�/

er D cos � iC sin � j, e� D � sin � iC cos � j; ez D k:

© Springer International Publishing Switzerland 2015S. Salsa, G. Verzini, Partial Differential Equations in Action. Complements and Exercises,UNITEXT – La Matematica per il 3+2 87, DOI 10.1007/978-3-319-15416-9_B

414 Appendix B Identities

1. gradient:

rf D @f

@rer C 1

r

@f

@�e� C @f

@zez

2. divergence:

div F D1

r

@

@r.rFr /C 1

r

@

@�F� C @

@zFz

3. Laplacian:

�f D @2f

@r2C 1

r

@f

@rC 1

r2@2f

@�2C @2f

@z2D 1

r

@

@r

�r@f

@r

C 1

r2@2f

@�2C @2f

@z2

4. curl:

curl F D1

r

ˇ̌̌̌ˇ̌ er re� ez@r @� @zFr rF� Fz

ˇ̌̌̌ˇ̌

Spherical coordinates

x D r cos � sin ; y D r sin � sin ; z D r cos .r > 0, 0 � � � 2� , 0 � � �/

er D cos � sin iC sin � sin jC cos k

e� D � sin � iC cos � j

ez D cos � cos iC sin � cos j� sin k:

1. gradient:

rf D @f

@rer C 1

r sin

@f

@�e� C 1

r

@f

@ e

2. divergence:

div F D @

@rFr C 2

rFr„ ƒ‚ …

radial part

C 1

r

1

sin

@

@�F� C @

@ F C cot F

�„ ƒ‚ …

spherical part

3. Laplacian:

�f D @2f

@r2C 2

r

@f

@r„ ƒ‚ …radial part

C 1

r2

²1

.sin /2@2f

@�2C @2f

@ 2C cot

@f

@

³„ ƒ‚ …

spherical part (Laplace-Beltrami operator)

4. curl:

curl F D 1

r2 sin

ˇ̌̌̌ˇ̌ er re r sin e�@r @ @�Fr rF r sin Fz

ˇ̌̌̌ˇ̌ :

B.2 Formulas 415

B.2 Formulas

Gauss’s formulas

The following formulas hold on Rn, n � 2, and we denote by:

• � a bounded domain with regular boundary @� and outward normal �.

• u; v vector fields that are regular1 up to the boundary of �.

• '; regular scalar fields up to the boundary of �.

• d� the infinitesimal surface element of @�:

We have the following formulas:

1.R div u dx D R

@ u � � d� (divergence formula)

2.R r' dx D R

@ '� d�

3.R �' dx D R

@ r' � � d� D R@ @�' d�

4.R div F dx D R

@ F � � d� � R

r � F dx

5.R �' dx D R

@ @�' d� � R r' � r dx (integration by parts)

6.R . �' � '� / dx D R

@ . @�' �'@� / d�

7.R curl u dx D � R@ u � � d�

8.R u � curl v dx D R

- v� curl u dxC R@ .v � u/ � � d�

Identities

div curl u D 0

curl grad' D 0

div .'u/ D ' div uCr' � u

curl .'u/ D ' curl uCr' � u

curl.u � v/ D .v � r/u� .u � r/ vC .div v/ u� .div u/ v

div.u � v/ D curl u � v � curl v � u

r .u � v/ D u � curl v C v � curl u C .u � r/ vC .v � r/u

.u � r/u D curl u � uC 12r juj2

curl curl u D r.div u/ ��u .curl curl D grad div � Laplacian/:

1 C 1���

is enough.

416 Appendix B Identities

B.3 Fourier Transforms

bu .�/ DZ

Ru .x/ e�i�x dx

General formulas

u buu .x � a/ e�ia�bu .�/eiaxu .x/ bu .� � a/

u .ax/ , a > 01

abu� �

a

u0 .x/ i�bu .�/xu .x/ ibu0 .�/.u � v/ .x/ bu .�/bv .�/u .x/ v .x/ .bu �bv/ .�/

Special transforms

u bue�ajxj, a > 0

2a

a2 C �2

1

a2 C x2�

ae�aj�j

e�ax2, a > 0

r�

ae� �2

4a

sin x

xe�jxj arctan

2

�2

�Œ�a;a� .x/ 2sin a�

�ı .x/ 1

1 2�ı .�/

B.4 Laplace Transforms 417

B.4 Laplace Transforms

eu .s/ DZ C1

0

u .t/ e�st dt

General formulas (u.t/ D 0 for t < 0)

u euu .t � a/ , a > 0 e�aseu .s/eatu .t/ , a 2 C eu .s � a/u .at/ , a > 0

1

aeu � s

a

�u0 .t/ seu .s/ � u.0C/u00 .t/ s2eu .s/ � u0.0C/ � su.0C/tu .t/ �eu0 .s/u .t/

t

R C1s

eu .�/ d�R t0u .�/ d�

eu .s/s

.u � v/ .t/ eu .s/ev .s/Special transforms

u euH .t/eat , a 2 C

1

s � aH .t/ sin at , a 2 R

a

s2 C a2

H .t/ cos at , a 2 Rs

s2 C a2

H .t/ sinh at , a 2 Ra

s2 � a2H .t/ cosh at , a 2 R

s

s2 � a2H .t/tn, n 2 N

nŠ

snC1

H .t/t˛, Re˛ > �1 �.˛ C 1/

s˛C1H .t/e�t2 es

2=4R C1s=2

e��2d�

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Collana Unitext – La Matematica per il 3+2

Series Editors:A. Quarteroni (Editor-in-Chief)L. AmbrosioP. BiscariC. CilibertoM. LedouxW.J. Runggaldier

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As of 2004, the books published in the series have been given a volume number.Titles in grey indicate editions out of print.As of 2011, the series also publishes books in English.

A. Bernasconi, B. CodenottiIntroduzione alla complessità computazionale1998, X+260 pp, ISBN 88-470-0020-3

A. Bernasconi, B. Codenotti, G. RestaMetodi matematici in complessità computazionale1999, X+364 pp, ISBN 88-470-0060-2

E. Salinelli, F. TomarelliModelli dinamici discreti2002, XII+354 pp, ISBN 88-470-0187-0

S. BoschAlgebra2003, VIII+380 pp, ISBN 88-470-0221-4

S. Graffi, M. Degli EspostiFisica matematica discreta2003, X+248 pp, ISBN 88-470-0212-5

S. Margarita, E. SalinelliMultiMath – Matematica Multimediale per l’Università2004, XX+270 pp, ISBN 88-470-0228-1

A. Quarteroni, R. Sacco, F.SaleriMatematica numerica (2a Ed.)2000, XIV+448 pp, ISBN 88-470-0077-72002, 2004 ristampa riveduta e corretta(1a edizione 1998, ISBN 88-470-0010-6)

13. A. Quarteroni, F. SaleriIntroduzione al Calcolo Scientifico (2a Ed.)2004, X+262 pp, ISBN 88-470-0256-7(1a edizione 2002, ISBN 88-470-0149-8)

14. S. SalsaEquazioni a derivate parziali - Metodi, modelli e applicazioni2004, XII+426 pp, ISBN 88-470-0259-1

15. G. RiccardiCalcolo differenziale ed integrale2004, XII+314 pp, ISBN 88-470-0285-0

16. M. ImpedovoMatematica generale con il calcolatore2005, X+526 pp, ISBN 88-470-0258-3

17. L. Formaggia, F. Saleri, A. VenezianiApplicazioni ed esercizi di modellistica numericaper problemi differenziali2005, VIII+396 pp, ISBN 88-470-0257-5

18. S. Salsa, G. VerziniEquazioni a derivate parziali – Complementi ed esercizi2005, VIII+406 pp, ISBN 88-470-0260-52007, ristampa con modifiche

19. C. Canuto, A. TabaccoAnalisi Matematica I (2a Ed.)2005, XII+448 pp, ISBN 88-470-0337-7(1a edizione, 2003, XII+376 pp, ISBN 88-470-0220-6)

20. F. Biagini, M. CampaninoElementi di Probabilità e Statistica2006, XII+236 pp, ISBN 88-470-0330-X

21. S. Leonesi, C. ToffaloriNumeri e Crittografia2006, VIII+178 pp, ISBN 88-470-0331-8

22. A. Quarteroni, F. SaleriIntroduzione al Calcolo Scientifico (3a Ed.)2006, X+306 pp, ISBN 88-470-0480-2

23. S. Leonesi, C. ToffaloriUn invito all’Algebra2006, XVII+432 pp, ISBN 88-470-0313-X

24. W.M. Baldoni, C. Ciliberto, G.M. Piacentini CattaneoAritmetica, Crittografia e Codici2006, XVI+518 pp, ISBN 88-470-0455-1

25. A. QuarteroniModellistica numerica per problemi differenziali (3a Ed.)2006, XIV+452 pp, ISBN 88-470-0493-4(1a edizione 2000, ISBN 88-470-0108-0)(2a edizione 2003, ISBN 88-470-0203-6)

26. M. Abate, F. TovenaCurve e superfici2006, XIV+394 pp, ISBN 88-470-0535-3

27. L. GiuzziCodici correttori2006, XVI+402 pp, ISBN 88-470-0539-6

28. L. RobbianoAlgebra lineare2007, XVI+210 pp, ISBN 88-470-0446-2

29. E. Rosazza Gianin, C. SgarraEsercizi di finanza matematica2007, X+184 pp, ISBN 978-88-470-0610-2

30. A. MachìGruppi – Una introduzione a idee e metodi della Teoria dei Gruppi2007, XII+350 pp, ISBN 978-88-470-0622-52010, ristampa con modifiche

31 Y. Biollay, A. Chaabouni, J. StubbeMatematica si parte!A cura di A. Quarteroni2007, XII+196 pp, ISBN 978-88-470-0675-1

32. M. ManettiTopologia2008, XII+298 pp, ISBN 978-88-470-0756-7

33. A. PascucciCalcolo stocastico per la finanza2008, XVI+518 pp, ISBN 978-88-470-0600-3

34. A. Quarteroni, R. Sacco, F. SaleriMatematica numerica (3a Ed.)2008, XVI+510 pp, ISBN 978-88-470-0782-6

35. P. Cannarsa, T. D’AprileIntroduzione alla teoria della misura e all’analisi funzionale2008, XII+268 pp, ISBN 978-88-470-0701-7

36. A. Quarteroni, F. SaleriCalcolo scientifico (4a Ed.)2008, XIV+358 pp, ISBN 978-88-470-0837-3

37. C. Canuto, A. TabaccoAnalisi Matematica I (3a Ed.)2008, XIV+452 pp, ISBN 978-88-470-0871-3

38. S. GabelliTeoria delle Equazioni e Teoria di Galois2008, XVI+410 pp, ISBN 978-88-470-0618-8

39. A. QuarteroniModellistica numerica per problemi differenziali (4a Ed.)2008, XVI+560 pp, ISBN 978-88-470-0841-0

40. C. Canuto, A. TabaccoAnalisi Matematica II2008, XVI+536 pp, ISBN 978-88-470-0873-12010, ristampa con modifiche

41. E. Salinelli, F. TomarelliModelli Dinamici Discreti (2a Ed.)2009, XIV+382 pp, ISBN 978-88-470-1075-8

42. S. Salsa, F.M.G. Vegni, A. Zaretti, P. ZuninoInvito alle equazioni a derivate parziali2009, XIV+440 pp, ISBN 978-88-470-1179-3

43. S. Dulli, S. Furini, E. PeronData mining2009, XIV+178 pp, ISBN 978-88-470-1162-5

44. A. Pascucci, W.J. RunggaldierFinanza Matematica2009, X+264 pp, ISBN 978-88-470-1441-1

45. S. SalsaEquazioni a derivate parziali – Metodi, modelli e applicazioni (2a Ed.)2010, XVI+614 pp, ISBN 978-88-470-1645-3

46. C. D’Angelo, A. QuarteroniMatematica Numerica – Esercizi, Laboratori e Progetti2010, VIII+374 pp, ISBN 978-88-470-1639-2

47. V. MorettiTeoria Spettrale e Meccanica Quantistica – Operatori in spazi di Hilbert2010, XVI+704 pp, ISBN 978-88-470-1610-1

48. C. Parenti, A. ParmeggianiAlgebra lineare ed equazioni differenziali ordinarie2010, VIII+208 pp, ISBN 978-88-470-1787-0

49. B. Korte, J. VygenOttimizzazione Combinatoria. Teoria e Algoritmi2010, XVI+662 pp, ISBN 978-88-470-1522-7

50. D. MundiciLogica: Metodo Breve2011, XII+126 pp, ISBN 978-88-470-1883-9

51. E. Fortuna, R. Frigerio, R. PardiniGeometria proiettiva. Problemi risolti e richiami di teoria2011, VIII+274 pp, ISBN 978-88-470-1746-7

52. C. PresillaElementi di Analisi Complessa. Funzioni di una variabile2011, XII+324 pp, ISBN 978-88-470-1829-7

53. L. Grippo, M. SciandroneMetodi di ottimizzazione non vincolata2011, XIV+614 pp, ISBN 978-88-470-1793-1

54. M. Abate, F. TovenaGeometria Differenziale2011, XIV+466 pp, ISBN 978-88-470-1919-5

55. M. Abate, F. TovenaCurves and Surfaces2011, XIV+390 pp, ISBN 978-88-470-1940-9

56. A. AmbrosettiAppunti sulle equazioni differenziali ordinarie2011, X+114 pp, ISBN 978-88-470-2393-2

57. L. Formaggia, F. Saleri, A. VenezianiSolving Numerical PDEs: Problems, Applications, Exercises2011, X+434 pp, ISBN 978-88-470-2411-3

58. A. MachìGroups. An Introduction to Ideas and Methods of the Theory of Groups2011, XIV+372 pp, ISBN 978-88-470-2420-5

59. A. Pascucci, W.J. RunggaldierFinancial Mathematics. Theory and Problems for Multi-period Models2011, X+288 pp, ISBN 978-88-470-2537-0

60. D. MundiciLogic: a Brief Course2012, XII+124 pp, ISBN 978-88-470-2360-4

61. A. MachìAlgebra for Symbolic Computation2012, VIII+174 pp, ISBN 978-88-470-2396-3

62. A. Quarteroni, F. Saleri, P. GervasioCalcolo Scientifico (5a ed.)2012, XVIII+450 pp, ISBN 978-88-470-2744-2

63. A. QuarteroniModellistica Numerica per Problemi Differenziali (5a ed.)2012, XVIII+628 pp, ISBN 978-88-470-2747-3

64. V. MorettiSpectral Theory and QuantumMechanicsWith an Introduction to the Algebraic Formulation2013, XVI+728 pp, ISBN 978-88-470-2834-0

65. S. Salsa, F.M.G. Vegni, A. Zaretti, P. ZuninoA Primer on PDEs. Models, Methods, Simulations2013, XIV+482 pp, ISBN 978-88-470-2861-6

66. V.I. ArnoldReal Algebraic Geometry2013, X+110 pp, ISBN 978-3-642–36242-2

67. F. Caravenna, P. Dai PraProbabilità. Un’introduzione attraverso modelli e applicazioni2013, X+396 pp, ISBN 978-88-470-2594-3

68. A. de Luca, F. D’AlessandroTeoria degli Automi Finiti2013, XII+316 pp, ISBN 978-88-470-5473-8

69. P. Biscari, T. Ruggeri, G. Saccomandi, M. VianelloMeccanica Razionale2013, XII+352 pp, ISBN 978-88-470-5696-3

70. E. Rosazza Gianin, C. SgarraMathematical Finance: Theory Review and Exercises. From BinomialModel to Risk Measures2013, X+278pp, ISBN 978-3-319-01356-5

71. E. Salinelli, F. TomarelliModelli Dinamici Discreti (3a Ed.)2014, XVI+394pp, ISBN 978-88-470-5503-2

72. C. PresillaElementi di Analisi Complessa. Funzioni di una variabile (2a Ed.)2014, XII+360pp, ISBN 978-88-470-5500-1

73. S. Ahmad, A. AmbrosettiA Textbook on Ordinary Differential Equations2014, XIV+324pp, ISBN 978-3-319-02128-7

74. A. Bermúdez, D. Gómez, P. SalgadoMathematical Models and Numerical Simulation in Electromagnetism2014, XVIII+430pp, ISBN 978-3-319-02948-1

75. A. QuarteroniMatematica Numerica. Esercizi, Laboratori e Progetti (2a Ed.)2013, XVIII+406pp, ISBN 978-88-470-5540-7

76. E. Salinelli, F. TomarelliDiscrete Dynamical Models2014, XVI+386pp, ISBN 978-3-319-02290-1

77. A. Quarteroni, R. Sacco, F. Saleri, P. GervasioMatematica Numerica (4a Ed.)2014, XVIII+532pp, ISBN 978-88-470-5643-5

78. M. ManettiTopologia (2a Ed.)2014, XII+334pp, ISBN 978-88-470-5661-9

79. M. Iannelli, A. PuglieseAn Introduction to Mathematical Population Dynamics.Along the trail of Volterra and Lotka2014, XIV+338pp, ISBN 978-3-319-03025-8

80. V.M. Abrusci, L. Tortora de FalcoLogica. Volume 12014, X+180pp, ISBN 978-88-470-5537-7

81. P. Biscari, T. Ruggeri, G. Saccomandi, M. VianelloMeccanica Razionale (2a Ed.)2014, XII+390pp, ISBN 978-88-470-5725-8

82. C. Canuto, A. TabaccoAnalisi Matematica I (4a Ed.)2014, XIV+508pp, ISBN 978-88-470-5722-7

83. C. Canuto, A. TabaccoAnalisi Matematica II (2a Ed.)2014, XII+576pp, ISBN 978-88-470-5728-9

84. C. Canuto, A. TabaccoMathematical Analysis I (2nd Ed.)2015, XIV+484pp, ISBN 978-3-319-12771-2

85. C. Canuto, A. TabaccoMathematical Analysis II (2nd Ed.)2015, XII+550pp, ISBN 978-3-319-12756-9

86. S. SalsaPartial Differential Equations in Action. FromModelling to Theory (2nd Ed.)2015, XVIII+688, ISBN 978-3-319-15092-5

87. S. Salsa, G. VerziniPartial Differential Equations in Action. Complements and Exercises2015, VIII+422, ISBN 978-3-319-15415-2

The online version of the books published in this series is available atSpringerLink.For further information, please visit the following link:http://www.springer.com/series/5418