Section: 17 | Special Functions |
Help Manual

Page of 1
Type a page number and hit Enter.
/1
  Back to Search Results
Type a page number and hit Enter.
Additional Information
Summary of table differences
No records found.
How to Cite this Reference
The recommended form of citation is:
John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), CRC Press/Taylor & Francis, Boca Raton, FL.
If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

8 SPECIAL FUNCTIONS

8.1 ORTHOGONAL POLYNOMIALS

  1. Legendre  Symbol:Pn(x)   Interval: [ 1,1 ]

    Differential Equation:(1x2)y2xy+n(n+1)y=0

    Explicit Expression:Pn(x)=12nm=0[n/2](1)m(nm)(2n2mn)xn2m

    Recurrence Relation:(n+1)Pn+1(x)=(2n+1)xPn(x)nPn1(x)

    Weight:  1

    Standardization:Pn(1)=1

    Norm:1+1[Pn(x)]2dx=22n+1

    Rodrigues' Formula:Pn(x)=(1)n2nn!dndxn{(1x2)n}

    Generating Function:R1=n=0Pn(x)zn;1<x<1,   |z|   <1,R=12xz+z2

    Inequality:|Pn(x)|1,1x1 .

  2. Tschebysheff, First Kind  Symbol:Tn(x)   Interval:[−1, 1]

    Differential Equation:(1x2)yxy+n2y=0

    Explicit Expression:n2m=0[n/2](1)m(nm1)!m!(n2m)!(2x)n2m=cos(narccosx)=Tn(x)

    Recurrence Relation:Tn+1(x)=2xTn(x)Tn1(x)

    Weight:(1x2)1/2

    Standardization:Tn(1)=1

    Norm:1+1(1x2)1/2[Tn(x)]2dx={π/2,n0π,n=0

    Rodrigues' Formula:(1)n(1x2)1/2π2n+1Γ(n+12)dndxn{(1x2)n(1/2)}=Tn(x)

    Generating Function:1xz12xzz2=n=0Tn(x)zn,1<x<1,   |z|<1

    Inequality:|Tn(x)|1,1x1 .

  3. Tschebysheff, Second KindSymbolUn(x)   Interval: [−1, 1]

    Differential Equation:(1x2)y3xy+n(n+2)y=0

    Explicit Expression: Un(x)=m=0[n/2](1)m(mn)!m!(n2m)!(2x)n2mUn(cosθ)=sin[(n+1)θ]sinθ

    Recurrence Relation:Un+1(x)=2xUn(x)Un1(x)

    Weight:(1x2)1/2

    Standardization:Un(1)=n+1

    Norm:1+1(1x2)1/2[Un(x)]2dx=π2

    Rodrigues' Formula:Un(x)=(1)n(n+1)π(1x2)1/22n+1Γ(n+32)dndxn{(1x2)n+(1/2)}

    Generating Function:112xz+z2=n=0Un(x)zn,1<x<1,   |z|<1

    Inequality:|Un(x)|n+1,1x1.

  4. Jacobi  Symbol:Pn(α,β)(x)   Interval: [−1, 1]

    Differential Equation:(1x2)y+[βα(α+β+2)x]y+n(n+α+β+1)y=0

    Explicit Expression:Pn(α,β)(x)=12nm=0n(n+αm)(n+βnm)(x1)nm(x+1)m

    Recurrence Relation:2(n+1)(n+α+β+1)(2n+α+β)Pn+1(α,β)(x)   =(2n+α+β+1)[(α2β2)+(2n+α+β+2)   ×(2n+α+β)x]Pn(α,β)(x)      2(n+α)(n+β)(2n+α+β+2)Pn1(α,β)(x)

    Weight:(1x)α(1+x)β;α,β>1

    Standardization:Pn(α,β)(x)=n+αn

    Norm:1+1(1x)α(1+x)β[Pn(α,β)(x)]2dx=2α+β+1Γ(n+α+1)Γ(n+β+1)(2n+α+β+1)n!Γ(n+α+β+1)

    Rodrigues' Formula:Pn(α,β)(x)=(1)n2nn!(1x)α(1+x)βdndxn{(1x)n+α(1+x)n+β}

    Generating Function:R1(1z+R)α(1+z+R)β=n=02αβPn(α,β)(x)zn,R=12xz+z2,   |z|<1

    Inequality:max1x1|Pn(α,β)(x)|={(n+qn)~nqifq=max(α,β)12|Pn(α,β)(x)|~n1/2ifq<12xis one of the two maximum points nearestβαα+β+1

  5. Generalized Laguerre  Symbol:Ln(α)(x)   Interval:[0,]

    Differential Equation:xy+(α+1x)y+ny=0

    Explicit Expression:Ln(α)(x)=m=0n(1)m(n+αnm)1m!xm

    Recurrence Relation:(n+1)Ln(α)+1(x)=[(2n+α+1)x]Ln(α)(x)(n+α)Ln(α)1(x)

    Weight:xαex,α>1

    Standardization:Ln(α)(x)=(1)nn!xn+

    Norm:0xαex[Ln(α)(x)]2dx=Γ(n+α+1)n!

    Rodrigues' Formula:Ln(α)(x)=1n!xαexdndxn{xn+αex}

    Generating Function:(1z)α1exp(xzz1)=n=0Ln(α)(x)zn

    Inequality:Ln(α)(x)Γ(n+α+1)n!Γ(α+1)ex/2;   x0α>0|Ln(a)(x)|[2Γ(α+n+1)n!Γ(α+1)]ex/2;   x01<α<0

  6. Hermite  Symbol:Hn(x)   Interval:[,]

    Differential Equation:y2xy+2ny=0

    Explicit Expression:Hn(x)=m=0[n/2](1)mn!(2x)n2mm!(n2m)!

    Recurrence Relation:Hn+1(x)=2xHn(x)2nHn1(x)

    Weight:ex2

    Standardization:Hn(1)=2nxn+

    Norm:ex2[Hn(x)]2dx=2nn!π

    Rodrigues' Formula:Hn(x)=(1)nex2dndxn(ex2)

    Generating Function:ex2+2zx=n=0Hn(x)znn!

    Inequality:|Hn(x)|ex2/2k2n/2n!k1.086435

8.2 TABLES OF ORTHOGONAL POLYNOMIALS

In the following, {Hn,Ln,Pn,Tn,Un} represent the nth order Hermite, Laguerre, Legendre, Tschebysheff (first kind), and Tschebysheff (second kind) polynomials.

H0=1 x10=(30240H0+75600H2+25200H4+2520H6+90H8+H10)/1024
H1=2x x9=(15120H1+10080H3+1512H5+72H7+H9)/512
H2=4x22 x8=(1680H0+3360H2+840H4+56H6+H8)/256
H3=8x312x x7=(840H1+420H3+42H5+H7)/128
H4=16x448x2+12 x6=(120H0+180H2+30H4+H6)/64
H5=32x5160x3+120x x5=(60H1+20H3+H5)/32
H6=64x6480x4+720x2120 x4=(12H0+12H2+H4)/16
H7=128x71344x5+3360x31680x x3=(6H1+H3)/8
H8=256x83584x6+13440x413440x2+1680 x2=(2H0+H2)/4
H9=512x99216x7+48384x580640x3+30240x x=(H1)/2
H10=1024x1023040x8+161280x6403200x4+302400x230240 1=H0
L0=1 x6=720L04320L1+10800L214400L3+10800L44320L5+720L6
L1=x+1 x5=120L0600L1+1200L21200L3+600L4120L5
L2=(x24x+2)/2 x4=24L096L1+144L296L3+24L4
L3=(x3+9x218x+6)/6 x3=6L018L1+18L26L3
L4=(x416x3+72x296x+24)/24 x2=2L04L1+2L2
L5=(x5+25x4200x3+600x2600x+120)/120 x=L0L1
L6=(x636x5+450x42400x3+5400x24320x+720)/720 1=L0
P0=1 x10=(4199P0+16150P2+15504P4+7904P6+2176P8+256P10)/46189
P1=x x9=(3315P1+4760P3+2992P5+960P7+128P9)/12155
P2=(3x21)/2 x8=(715P0+2600P2+2160P4+832P6+128P8)/6435
P3=(5x33x)/2 x7=(143P1+182P3+88P5+16P7)/429
P4=(35x430x2+3)/8 x6=(33P0+110P2+72P4+16P6)/231
P5=(63x570x3+15x)/8 x5=(27P1+28P3+8P5)/63
P6=(231x6315x4+105x25)/16 x4=(7P0+20P2+8P4)/35
P7=(429x7693x5+315x335x)/16 x3=(3P1+2P3)/5
P8=(6435x812012x6+6930x41260x2+35)/128 x2=(P0+2P2)/3
P9=(12155x925740x7+18018x54620x3+315x)/128 x=P1
P10=(46189x10109395x8+90090x630030x4+3465x263)/256 1=P0
T0=1 x10=(126T0+210T2+120T4+45T6+10T8+T10)/512
T1=x x9=(126T1+84T3+36T5+9T7+T9)/256
T2=2x21 x8=(35T0+56T2+28T4+8T6+T8)/128
T3=4x33x x7=(35T1+21T3+7T5+T7)/64
T4=8x48x2+1 x6=(10T0+15T2+6T4+T6)/32
T5=16x520x3+5x x5=(10T1+5T3+T5)/16
T6=32x648x4+18x21 x4=(3T0+4T2+T4)/8
T7=64x7112x5+56x37x x3=(3T1+T3)/4
T8=128x8256x6+160x432x2+1 x2=(T0+T2)/2
T9=256x9576x7+432x5120x3+9x x=T1
T10=512x101280x8+1120x6400x4+50x21 1=T0
U0=1 x10=(42U0+90U2+75U4+35U6+9U8+U10)/1024
U1=2x x9=(42U1+48U3+27U5+8U7+U9)/512
U2=4x21 x8=(14U0+28U2+20U4+7U6+U8)/256
U3=8x34x x7=(14U1+14U3+6U5+U7)/128
U4=16x412x2+1 x6=(5U0+9U2+5U4+U6)/64
U5=32x532x3+6x x5=(5U1+4U3+U5)/32
U6=64x680x4+24x21 x4=(2U0+3U2+U4)/16
U7=128x7192x5+80x38x x3=(2U1+U3)/8
U8=256x8448x6+240x440x2+1 x2=(U0+U2)/4
U9=512x91024x7+672x5160x3+10x x=(U1)/2
U10=1024x102304x8+1792x6560x4+60x21 1=U0

8.3 BESSEL FUNCTIONS

  1. Bessel's differential equation for a real variable x is x2d2ydx2+xdydx+(x2n2)y=0
  2. When n is not an integer, two independent solutions of the equation are Jn(x) and Jn(x) where Jn(x)=∑k=0∞(−1)kk!Γ(n+k+1)(x2)n+2k
  3. If n is an integer then Jn(x)=(1)nJn(x) , where Jn(x)=xn2nn!{1x222·1!(n+1)+x424·2!(n+1)(n+2)+x626·3!(n+1)(n+2)(n+3)+}
  4. For n=0 and n=1 , this formula becomes J0(x)=1x222(1!)2+x424(2!)2x626(3!)2+x828(4!)2J1(x)=x2x323·1!2!+x525·2!3!x727·3!4!+x929·4!5!
  5. Table of zeros for J0(x) and J1(x) . Define {αn,βn} by J0(αn)=0 and J1(βn)=0 .
    Roots αn J1(αn) Roots βn J0(βn)
    2.4048 0.5191 0.0000 1.0000
    5.5201 0.3403 3.8317 0.4028
    8.6537 0.2715 7.0156 0.3001
    11.7915 0.2325 10.1735 0.2497
    14.9309 0.2065 13.3237 0.2184
    18.0711 0.1877 16.4706 0.1965
    21.2116 0.1733 19.6159 0.1801
  6. Recurrence formulas Jn1(x)+Jn+1(x)=2nxJn(x)nJn(x)+xJn(x)=xJn1(x)Jn1(x)Jn+1(x)=2Jn(x)nJn(x)xJn(x)=xJn+1(x)
  7. If Jn is written for Jn(x) and Jn(k) is written for dkdxk{Jn(x)} , then the following derivative relationships are important J0(r)=J1(r1)J0(2)=J0+1xJ1=12(J2J0)J0(3)=1xJ0+(12x2)J1=14(J3+3J1)J0(4)=(13x2)J0(2x6x3)J1=18(J44J2+3J0),etc.
  8. Half-order Bessel functions J12(x)=2πxsinxJ12(x)=2πxcosxJn+32(x)=xn+12ddx{x(n+12)Jn+12(x)}Jn12(x)=x(n+12)ddx{xn+12Jn+12(x)}
    n (πx2)12Jn+12(x) (πx2)12J(n+12)(x)
    0 sinx cosx
    1 sinxxcosx cosxxsinx
    2 (3x21)sinx3xcosx (3x21)cosx+3xsinx
    3 (15x36x)sinx(15x21)cosx (15x36x)cosx(15x21)sinx
  9. Additional solutions to Bessel's equation are Yn(x) (also called Weber's function, and sometimes denoted by Nn(x) ) and Hn(1)(x) and Hn(2)(x) (also called Hankel functions) These solutions are defined as follows Yn(x)={Jn(x)cos(nπ)−J−n(x)sin(nπ)nnot an integerHn(1)(x)=Jn(x)+iYn(x)limv→nJv(x)cos(vπ)−J−v(x)sin(vπ)nan integerHn(2)(x)=Jn(x)−iYn(x) The additional properties of these functions may all be derived from the above relations and the known properties of Jn(x) .
  10. Complete solutions to Bessel's equation may be written as c1Jn(x)+c2Jn(x) when n is not an integer or, for any value of n , c1Jn(x)+c2Yn(x) or c1Hn(1)x+c2Hn(2)(x) .
  11. The modified (or hyperbolic) Bessel's differential equation is x2d2ydx2+xdydx(x2+n2)y=0
  12. When n is not an integer, two independent solutions of the equation are In(x) and In(x) , where In(x)=k=01k!Γ(n+k+1)(x2)n+2k
  13. If n is an integer, In(x)=In(x)=xn2nn!(1+x222·1!(n+1)+x424·2!(n+1)(n+2)+x626·3!(n+1)(n+2)(n+3)+)
  14. For n=0 and n=1 , this formula becomes I0(x)=1+x222(1!)2+x424(2!)2+x626(3!)2+x828(4!)2+I1(x)=x2+x323·1!2!+x525·2!3!+x727·3!4!+x929·4!5!+
  15. Another solution to the modified Bessel's equation is Kn(x)={12πI−n(x)−In(x)sin(nπ)nnot an integerlimv→n12πI−v(x)−Iv(x)sin(vπ)nan integer This function is linearly independent of In(x) for all values of n . Thus the complete solution to the modified Bessel's equation may be written as c1In(x)+c2In(x)   whennis not an integer or c1In(x)+c2Kn(x)   for any value ofn
  16. The following relations hold among the various Bessel functions: In(z)=imJm(iz)Yn(iz)=(i)n+1In(z)2πinKn(z) Most of the properties of the modified Bessel function may be deduced from the known properties of Jn(x) by use of these relations and those previously given.
  17. Recurrence formulas In1(x)In+1(x)=2nxIn(x)In1(x)+In+1(x)=2In(x)In1(x)nxIn(x)=In(x)In(x)=In+1(x)+nxIn(z)

8.4 FACTORIAL FUNCTION

For non-negative integers n , the factorial of n , denoted n! , is the product of all positive integers less than or equal to n ; n!=n·(n1)·(n2)2·1 . If n is a negative integer ( n=1,2, ) then n!=± .

Approximations to n! for large n include Stirling's formula n!2πe(ne)n+12, and Burnsides's formula n!2π(n+12e)n+12.

n n! log10n! n n! log10n!
0 1 0.00000 1 1 0.00000
2 2 0.30103 3 6 0.77815
4 24 1.38021 5 120 2.07918
6 720 2.85733 7 5040 3.70243
8 40320 4.60552 9 3.6288×105 5.55976
10 3.6288×106 6.55976 11 3.9917×107 7.60116
12 4.7900×108 8.68034 13 6.2270×109 9.79428
14 8.7178×1010 10.94041 15 1.3077×1012 12.11650
16 2.0923×1013 13.32062 17 3.5569×1014 14.55107
18 6.4024×1015 15.80634 19 1.2165×1017 17.08509
20 2.4329×1018 18.38612 25 1.5511×1025 25.19065
30 2.6525×1032 32.42366 40 8.1592×1047 47.91165
50 3.0414×1064 64.48307 60 8.3210×1081 81.92017
70 1.1979×10100 100.07841 80 7.1569×10118 118.85473
90 1.4857×10138 138.17194 100 9.3326×10157 157.97000
110 1.5882×10178 178.20092 120 6.6895×10198 198.82539
130 6.4669×10219 219.81069 150 5.7134×10262 262.75689
500 1.2201×101134 1134.0864 1000 4.0239×102567 2567.6046

8.5 Gamma Function

Definition:Γ(n)=0tn1etdt   n>0

Recursion Formula:Γ(n+1)=nΓ(n)Γ(n+1)=n!ifn=0,1,2,where0!=1For n<0the gamma function can be defined by using Γ(n)=Γ(n+1)n

Special Values:Γ(1/2)=πΓ(m+12)=1·3·5(2m1)2mπ   m=1,2,3,Γ(m+12)=(1)m2mπ1·3·5(2m1)      m=1,2,3,

Special Formulas:Γ(x+1)=limk1·2·3k(x+1)(x+2)(x+k)kx1Γ(x)=xeγxΠm=1{(1+xm)ex/m}      (γis Euler"s constant)

Properties:Γ(1)=0eγxlnxdx=γΓ(x)Γ(x)=γ+(111x)+(121x+1)++(1n1x+n1)+Γ(x+1)=2πxxxex{1+112x+1288x213951,840x3+}      (Stirling"s asymptotic series)

n Γ(n) n Γ(n) n Γ(n) n Γ(n)
1.00 1.00000 1.25 .90640 1.50 .88623 1.75 .91906
1.01 .99433 1.26 .90440 1.51 .88659 1.76 .92137
1.02 .98884 1.27 .90250 1.52 .88704 1.77 .92376
1.03 .98355 1.28 .90072 1.53 .88757 1.78 .92623
1.04 .97844 1.29 .89904 1.54 .88818 1.79 .92877
1.05 .97350 1.30 .89747 1.55 .88887 1.80 .93138
1.06 .96874 1.31 .89600 1.56 .88964 1.81 .93408
1.07 .96415 1.32 .89464 1.57 .89049 1.82 .93685
1.08 .95973 1.33 .89338 1.58 .89142 1.83 .93969
1.09 .95546 1.34 .89222 1.59 .89243 1.84 .94261
1.10 .95135 1.35 .89115 1.60 .89352 1.85 .94561
1.11 .94740 1.36 .89018 1.61 .89468 1.86 .94869
1.12 .94359 1.37 .88931 1.62 .89592 1.87 .95184
1.13 .93993 1.38 .88854 1.63 .89724 1.88 .95507
1.14 .93642 1.39 .88785 1.64 .89864 1.89 .95838
1.15 .93304 1.40 .88726 1.65 .90012 1.90 .96177
1.16 .92980 1.41 .88676 1.66 .90167 1.91 .96523
1.17 .92670 1.42 .88636 1.67 .90330 1.92 .96877
1.18 .92373 1.43 .88604 1.68 .90500 1.93 .97240
1.19 .92089 1.44 .88581 1.69 .90678 1.94 .97610
1.20 .91817 1.45 .88566 1.70 .90864 1.95 .97988
1.21 .91558 1.46 .88560 1.71 .91057 1.96 .98374
1.22 .91311 1.47 .88563 1.72 .91258 1.97 .98768
1.23 .91075 1.48 .88575 1.73 .91466 1.98 .99171
1.24 .90852 1.49 .88595 1.74 .91683 1.99 .99581
2.00 1.00000

8.6 BETA FUNCTION

Definition:B(m,n)=01tm1(1t)n1dt   m>0,n>0

Relationship with Gamma function:B(m,n)=Γ(m)Γ(n)Γ(m+n)

Properties:B(m,n)=B(n,m)B(m,n)=20π/2sin2m1θcos2n1θdθB(m,n)=0tm1(1+t)m+ndtB(m,n)=rn(r+1)m01tm1(1t)n1(r+t)m+ndt

8.7 ERROR FUNCTION

Definition:erf(x)=2π0xet2dt

Series:erf(x)=2π(xx33+12!x5513!x77+)

Property:erf(x)=erf(x)

Relationship with Normal Probability Functionf(t) :  0xf(t)dt=12erf(x2) To evaluate erf(2.3) , one proceeds as follows: For x2=2.3 , one finds x=(2.3)(2)=3.25 . In the normal probability function table, one finds the entry 0.4994 opposite the value 3.25. Thus erf(2.3)=2(0.4994)=0.9988 . erfc(z)=1erf(z)=2πzet2dt is known as the complementary error function.

Page 1 of 1
1/1

Entry Display
This is where the entry will be displayed

Log In - Individual User
You are not within the network of a subscribing institution.
Please sign in with an Individual User account to continue.
Note that Workspace accounts are not valid.

Confirm Log Out
Are you sure?
Log In to Your Workspace
Your personal workspace allows you to save and access your searches and bookmarks.
Username
Password
Remember Me
This will save a cookie on your browser



If you do not have a workspace Log In click here to create one.
Forgotten your workspace password? Click here for an e-mail reminder.
Log Out From Your Workspace
Are you sure?
Create your personal workspace
Title
First Name (Given)
Last Name (Family)
Email address
Username
Password
Confirm Password


Incorrect login details
You have entered your Workspace sign in credentials instead of Individual User sign in credentials.
You must be authenticated within your organisation's network IP range in order to access your Workspace account.
Click the help icon for more information on the differences between these two accounts.
Incorrect login details
You have entered your Individual User account sign in credentials instead of Workspace credentials.
While using this network, a personal workspace account can be created to save your bookmarks and search preferences for later use.
Click the help icon for more information on the differences between Individual User accounts and Workspace accounts.
My Account

Change Your Workspace Password
Username
Current Password

New Password
Confirm New Password


Update your Personal Workspace Details
Username
Title
First Name (Given)
Last Name (Family)
Email address


Workspace Log In Reminder
Please enter your username and/or your e-mail address:

Username
Email Address


Searching for Chemicals and Properties

The CRC Handbook of Chemistry and Physics (HBCP) contains over 700 tables in over 450 documents which may be divided into several pages, all categorised into 17 major subject areas. The search on this page works by searching the content of each page individually, much like any web search. This provides a challenge if you want to search for multiple terms and those terms exist on different pages, or if you use a synonym/abbreviation that does not exist in the document.

We use metadata to avoid some of these issues by including certain keywords invisibly behind each table. Whilst this approach works well in many situations, like any web search it relies in the terms you have entered existing in the document with the same spelling, abbreviation etc.

Since chemical compounds and their properties are immutable, a single centralised database has been created from all chemical compounds throughout HBCP. This database contains every chemical compound and over 20 of the most common physical properties collated from each of the >700 tables. What's more, the properties can be searched numerically, including range searching, and you can even search by drawing a chemical structure. A complete list of every document table in which the compound occurs is listed, and are hyperlinked to the relevant document table.

The 'Search Chemicals' page can be found by clicking the flask icon in the navigation bar at the top of this page. For more detailed information on how to use the chemical search, including adding properties, saving searches, exporting search results and more, click the help icon in to top right of this page, next to the welcome login message.

Below is an example of a chemical entry, showing its structure, physical properties and document tables in which it appears.

image of an example chemical entry
We use cookies to improve your website experience. To learn about our use of cookies and how you can manage your cookie settings, please see our Cookie Policy. By continuing to use the website, you consent to our use of cookies.
Cookie Policy

Cookie Policy

We have developed this cookie policy (the “Cookie Policy”) in order to explain how we use cookies and similar technologies (together, “Cookies”) on this website (the “Website”) and to demonstrate our firm commitment to the privacy of your personal information.

The first time that you visit our Website, we notify you about our use of Cookies through a notification banner. By continuing to use the Website, you consent to our use of Cookies as described in this Cookie Policy. However, you can choose whether or not to continue accepting Cookies at any later time. Information on how to manage Cookies is set out later in this Cookie Policy.

Please note that our use of any personal information we collect about you is subject to our Privacy Policy.

What are Cookies?

Cookies are small text files containing user IDs that are automatically placed on your computer or other device by when you visit a website. The Cookies are stored by the internet browser. The browser sends the Cookies back to the website on each subsequent visit, allowing the website to recognise your computer or device. This recognition enables the website provider to observe your activity on the website, deliver a personalised, responsive service and improve the website.

Cookies can be ‘Session Cookies’ or ‘Persistent Cookies’. Session Cookies allow a website to link a series of your actions during one browser session, for example to remember the items you have added to a shopping basket. Session Cookies expire after a browser session and are therefore not stored on your computer or device afterwards. Persistent Cookies are stored on your computer or device between browser sessions and can be used when you make subsequent visits to the website, for example to remember your website preferences, such as language or font size.

Cookies We Use and Their Purpose

We use three types of Cookies - ‘Strictly Necessary’ Cookies, ‘Performance’ Cookies and ‘Functionality’ Cookies. Each type of Cookie and the purposes for which we use them are described in this section. To learn about the specific Cookies we use, please see our List of Cookies.

1. Strictly Necessary Cookies

‘Strictly Necessary’ Cookies enable you to move around the Website and use essential features. For example, if you log into the Website, we use a Cookie to keep you logged in and allow you to access restricted areas, without you having to repeatedly enter your login details. If you are registering for or purchasing a product or service, we will use Cookies to remember your information and selections, as you move through the registration or purchase process.

Strictly Necessary Cookies are necessary for our Website to provide you with a full service. If you disable them, certain essential features of the Website will not be available to you and the performance of the Website will be impeded.

2. Performance Cookies

‘Performance’ Cookies collect information about how you use our Website, for example which pages you visit and if you experience any errors. These Cookies don’t collect any information that could identify you – all the information collected is anonymous. We may use these Cookies to help us understand how you use the Website and assess how well the Website performs and how it could be improved.

3. Functionality Cookies

‘Functionality’ Cookies enable a website to provide you with specific services or a customised experience. We may use these Cookies to provide you with services such as watching a video or adding user comments. We may also use such Cookies to remember changes you make to your settings or preferences (for example, changes to text size or your choice of language or region) or offer you time-saving or personalised features.

You can control whether or not Functionality Cookies are used, but disabling them may mean we are unable to provide you with some services or features of the Website.

First and Third Party Cookies

The Cookies placed on your computer or device include ‘First Party’ Cookies, meaning Cookies that are placed there by us, or by third party service providers acting on our behalf. Where such Cookies are being managed by third parties, we only allow the third parties to use the Cookies for our purposes, as described in this Cookie Policy, and not for their own purposes.

The Cookies placed on your computer or device may also include ‘Third Party’ Cookies, meaning Cookies that are placed there by third parties. These Cookies may include third party advertisers who display adverts on our Website and/or social network providers who provide ‘like’ or ‘share’ capabilities (see the above section on Targeting or Advertising Cookies). They may also include third parties who provide video content which is embedded on our Website (such as YouTube). Please see the website terms and policies of these third parties for further information on their use of Cookies.

To learn about the specific First Party and Third Party Cookies used by our, please see our List of Cookies.

Managing Cookies

You always have a choice over whether or not to accept Cookies. When you first visit the Website and we notify you about our use of Cookies, you can choose not to consent to such use. If you continue to use the Website, you are consenting to our use of Cookies for the time being. However, you can choose not to continue accepting Cookies at any later time. In this section, we describe ways to manage Cookies, including how to disable them.

You can manage Cookies through the settings of your internet browser. You can choose to block or restrict Cookies from being placed on your computer or device. You can also review periodically review the Cookies that have been placed there and disable some or all of them.

You can learn more about how to manage Cookies on the following websites: www.allaboutcookies.org and www.youronlinechoices.com.

Please be aware that if you choose not to accept certain Cookies, it may mean we are unable to provide you with some services or features of the Website.

Changes to Cookie Policy

In order to keep up with changing legislation and best practice, we may revise this Cookie Policy at any time without notice by posting a revised version on this Website. Please check back periodically so that you are aware of any changes.

Questions or Concerns

If you have any questions or concerns about this Cookie Policy or our use of Cookies on the Website, please contact us by email to [email protected]

You can also contact the Privacy Officer for the Informa PLC group at [email protected].


Our Cookies

Here is a list of cookies we have defined as 'Strictly Necessary':

Taylor and Francis 'First Party' Cookies

JSESSIONID

TandF.ACCT.CNB.cookieId

TandF.WS.CNB.cookieId

TandF.SU.CNB.cookieId

TandF.PORTAL.cookiesAgreed

TandF.LOGIN.cookiesAgreed

TandF.HBCP.cookiesAgreed

TandF.CCD.cookiesAgreed

TandF.DNP.cookiesAgreed

TandF.DOC.cookiesAgreed

TandF.DOD.cookiesAgreed

TandF.DIOC.cookiesAgreed

TandF.POLY.cookiesAgreed

TandF.DFC.cookiesAgreed

TandF.DMNP.cookiesAgreed

TandF.DCCC.cookiesAgreed

TandF.POC.cookiesAgreed

Here is a list of the cookies we have defined as 'Performance'.

'Third Party' Cookies

Google Analytics:

_ga

_gid

_gat

Accessibility

The Voluntary Product Accessibility Template (VPAT) is a self-assessment document which discloses how accessible Information and Communication Technology products are in accordance with global standards.

The VPAT disclosure templates do not guarantee product accessibility but provide transparency around the product(s) and enables direction when accessing accessibility requirements.

Taylor & Francis has chosen to complete the International version of VPAT which encompasses Section 508 (US), EN 301 549 (EU) and WCAG2.1 (Web Content Accessibility Guidelines) for its products.

Click here for more information about how to use this web application using the keyboard.


This is replaced with text from the script
This is replaced with text from the script
Top Notification Bar Dialog Header
Your Session is about to Expire!
Your session will expire in seconds

Please move your cursor to continue.