Trigonometric Functions
sinz=eiz−e−iz2i,cosz=eiz+e−iz2,tanz=−i⋅eiz−e−izeiz+e−iz;
f(x)=sinx
f(x)=cosx
f(x)=tanx
f(x)=cscx
f(x)=secx
f(x)=cotx| θ | sin(θ) | cos(θ) | tan(θ) | csc(θ) | sec(θ) | cot(θ) |
| π6=30∘ | 12 | 3–√2 | 3–√3 | 2 | 23–√3 | 3–√ |
| π3=60∘ | 3–√2 | 12 | 3–√ | 23–√3 | 2 | 3–√3 |
| π4=45∘ | 2–√2 | 2–√2 | 1 | 2–√ | 2–√ | 1 |
| π12=15∘ | 6–√−2–√4 | 6–√+2–√4 | 2−3–√ | 6–√+2–√ | 6–√−2–√ | 2+3–√ |
| π8=22.5∘ | 2−2–√−−−−−−√2 | 2+2–√−−−−−−√2 | 2–√−1 | 4+22–√−−−−−−−√ | 4−22–√−−−−−−−√ | 2–√+1 |
| π10=18∘ | 5–√−14 | 10+25–√−−−−−−−−√4 | 5−25–√5−−−−−−−√ | 5–√+1 | 10−25–√5−−−−−−−−√ | 5+25–√−−−−−−−√ |
| π5=36∘ | 10−25–√−−−−−−−−√4 | 5–√+14 | 5−25–√−−−−−−−√ | 10+25–√5−−−−−−−−√ | 5–√−1 | 5+25–√5−−−−−−−√ |
sin(−x)=−sinx,cos(−x)=cosx,tan(−x)=−tanx;
sin(x±π)=−sinx,cos(x±π)=−cosx,tan(x±π)=tanx; sin(π−x)=sinx,cos(π−x)=−cosx,tan(π−x)=−tanx;
sin(x±π2)=±cosx,cos(x±π2)=∓sinx,tan(x±π2)=−cotx; sin(π2−x)=cosx,cos(π2−x)=sinx,tan(π2−x)=cotx;
sin(x±π4)=2–√2(sinx±cosx),cos(x±π4)=2–√2(cosx∓sinx),tan(x±π4)=sinx±cosxcosx∓sinx=tanx±11∓tanx;
sin(π4−x)=2–√2(cosx−sinx),cos(π4−x)=2–√2(cosx+sinx),tan(π4−x)=1−tanx1+tanx;
sin2x+cos2x=1,sec2x−tan2x=1,csc2x−cot2x=1;
sin(x±y)=sinxcosy±cosxsiny,cos(x±y)=cosxcosy∓sinxsiny,tan(x±y)=tanx±tany1∓tanxtany,cot(x±y)=cotxcoty∓1coty±cotx;
sin2x=2sinxcosx,cos2x=⎧⎩⎨cos2x−sin2x2cos2x−11−2sin2x,tan2x=2tanx1−tan2x,cot2x=cot2x−12cotx;
sin3x=⎧⎩⎨3sinx−4sin3x3sinxcos2x−sin3x4sinxcos2x−sinx,cos3x=⎧⎩⎨4cos3x−3cosxcos3x−3cosxsin2xcosx−4cosxsin2x,tan3x=3tanx−tan3x1−3tan2x;
sin(nx)=∑k=02k+1≤n(n2k+1)(−1)ksin2k+1xcosn−(2k+1)x, cos(nx)=∑k=02k≤n(n2k)(−1)ksin2kxcosn−2kx, tan(nx)=∑k=02k+1≤n(n2k+1)(−1)ktan2k+1x∑k=02k≤n(n2k)(−1)ktan2kx;
cos(nθ)+isin(nθ)=(cosθ+isinθ)n, [cos(nθ)sin(nθ)]=[cosθsinθ−sinθcosθ]n[10];
sin2x2=1−cosx2,cos2x2=1+cosx2,tan2x2=1−cosx1+cosx;sin2x=1−cos2x2,cos2x=1+cos2x2;tan2x=1−cos2x1+cos2x;
sinx⋅cosy=sin(x+y)+sin(x−y)2;sinx⋅siny=cos(x−y)−cos(x+y)2,cosx⋅cosy=cos(x+y)+cos(x−y)2;
sinx±siny=2sin(x±y2)cos(x∓y2);cosx+cosy=2cos(x+y2)cos(x−y2),cosx−cosy=−2sin(x+y2)sin(x−y2);
sinx+cosy=2sin(x+y+π22)sin(x−y+π22),sinx−cosy=−2cos(x+y+π22)cos(x−y+π22);
Inverse Trigonometric Functions
sin−1(z)=−i⋅ln(iz+1−z2−−−−−√),cos−1(z)=−i⋅ln(z±z2−1−−−−−√),tan−1(z)=12iln(1+iz1−iz),atan2(y,x)=12iln(x+yix−yi);
f(x)=sin−1x
f(x)=cos−1x
f(x)=tan−1x
f(x)=cot−1x
f(x)=sec−1x
f(x)=csc−1xsin−1(−x)=−sin−1(x),tan−1(−x)=−tan−1(x),csc−1(−x)=−csc−1(x),atan2(y,x)=−atan2(−y,x)≠−atan2(y,−x)=atan2(−y,−x);
csc−1x=sin−1(1x),sec−1x=cos−1(1x);tan−1x=cot−1(1x) (x>0);sin−1x+cos−1x=π2,tan−1x+cot−1x=π2,sec−1x+csc−1x=π2;
atan2(y,x)=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪tan−1(yx)tan−1(yx)+πtan−1(yx)−ππ2−π2x>0x<0, y≥0x<0, y<0x=0, y>0x=0, y<0=⎧⎩⎨⎪⎪⎪⎪⎪⎪cot−1(xy)cot−1(xy)−π0πy>0y<0y=0, x>0y=0, x<0=2tan−1(yx2+y2−−−−−−√+x)=2tan−1(x2+y2−−−−−−√−xy);
asin(kx)+bcos(kx)=a2+b2−−−−−−√cos(kx−atan2(a,b))=a2+b2−−−−−−√sin(kx+atan2(b,a));
sin−1(x)=cos−1(1−x2−−−−−√)=tan−1(x1−x2−−−−−√),cos−1(x)=sin−1(1−x2−−−−−√)=tan−1(1−x2−−−−−√x);(0<x<1)
sin(cos−1x)=1−x2−−−−−√,sin(tan−1x)=x1+x2−−−−−√,cos(sin−1x)=1−x2−−−−−√,cos(tan−1x)=11+x2−−−−−√,tan(cos−1x)=1−x2−−−−−√x,tan(sin−1x)=x1−x2−−−−−√;
cos(atan2(y,x))=xx2+y2−−−−−−√,sin(atan2(y,x))=yx2+y2−−−−−−√;tan−1(x)±tan−1(y)=atan2(x±y,1∓xy);
tan(atan2(y,x))=yx,cot(atan2(y,x))=xy;atan2(sinx,cosx)=x; (−π<x<π)
sin−1(sin(x))={x,sgn(x)π−x,|x|<π2π2<|x|<3π2,cos−1(sin(x))={π2−x,x−sgn(x)π+π2,|x|<π2π2<|x|<3π2,sin−1(cos(x))={π2−|x|,|x|−3π2,|x|<ππ<|x|<2π,cos−1(cos(x))={|x|,2π−|x|,|x|<ππ<|x|<2π;
tan−1(tan(x))=x−πround(xπ),tan−1(cot(x))=−x+π(12+floor(xπ)),cot−1(cot(x))=x−πfloor(xπ),cot−1(tan(x))=−x+π(12+round(xπ));
sin(2sin−1x)=2x1−x2−−−−−√,sin(2cos−1x)=2x1−x2−−−−−√;cos(2sin−1x)=1−2x2,cos(2cos−1x)=2x2−1;sin(2tan−1x)=2xx2+1,cos(2tan−1x)=1−x2x2+1,tan(2tan−1x)=2x1−x2;
sin(2atan2(y,x))=2xyx2+y2,cos(2atan2(y,x))=x2−y2x2+y2,tan(2atan2(y,x))=2xyx2−y2;
sin(3sin−1x)=3x−4x3,cos(3cos−1x)=4x3−3x;sin(3cos−1x)=(4x2−1)1−x2−−−−−√,cos(3sin−1x)=(1−4x2)1−x2−−−−−√;sin(3tan−1x)=3x−x3(x2+1)3/2,cos(3tan−1x)=1−3x2(x2+1)3/2,tan(3tan−1x)=x3−3x3x2−1;
sin(12sin−1(x))=12(1+x−−−−√−1−x−−−−√)=12–√⋅x1−x2−−−−−√+1−−−−−−−−−−√,cos(12sin−1(x))=12(1+x−−−−√+1−x−−−−√)=12–√⋅1−x2−−−−−√+1−−−−−−−−−−√,tan(12sin−1(x))=x1−x2−−−−−√+1;sin(12cos−1(x))=1−x−−−−√2–√,cos(12cos−1(x))=1+x−−−−√2–√,tan(12cos−1(x))=1−x−−−−√1+x−−−−√;sin(12tan−1(x))=12–√⋅x(1+x2)+1+x2−−−−−√−−−−−−−−−−−−−−−√,cos(12tan−1(x))=12–√⋅11+x2−−−−−√+1−−−−−−−−−−−√,tan(12tan−1(x))=x2+1−−−−−√−1x=xx2+1−−−−−√+1;sin(12atan2(y,x))=12–√⋅y(x2+y2)+xx2+y2−−−−−−√−−−−−−−−−−−−−−−−−−√,cos(12atan2(y,x))=12–√⋅xx2+y2−−−−−−√+1−−−−−−−−−−−√;
Trigonometric Equations
Solutions of asin(x)=b: x=arcsin(ba)+2πn and x=π−arcsin(ba)+2πn;
Solutions of acos(x)=b: x=±arccos(ba)+2πn;
Solutions of atan(x)=b: x=arctan(ba)+πn; Solutions of acos(x)+bsin(x)=0: x=arctan(−ab)+πn;
Solutions of acos(x)+bsin(x)+c=0: x=2arctan(b±a2+b2−c2−−−−−−−−−√a−c)+2πn;
Power, Exponential, and Logarithm
f(x)=ex
f(x)=lnxFor real numbers:
ax=exlna; a0=1, a−x=1ax, axy=ax−−√y;
axay=ax+y, axay=ax−y, (ax)y=axy, (ab)x=axbx, (ab)x=axbx;
x−−√=x1/2; x−−√y√=xy−−√, 1x−−√=1x−−√, x2−−√=|x|;
alogab=b, logab=logcblogca=lnblna; loga1=0, logaa=1;
logab+logac=logabc, logab−logac=loga(bc), −logbx=logb(1x);
logabk=k⋅logab, logakb=1klogab, logab⋅logbc=logac; a+logbx=logb(bax); logbx−−√=12logbx;
f(x)=x0.6
f(x)=x1.4
f(x)=x−0.6
f(x)=x−1.6
f(x)=1.6x
f(x)=0.4x
f(x)=log0.4x
f(x)=log1.8xexp(a+bi)=ea(cosb+isinb),ln(a+bi)=lna2+b2−−−−−−√+atan2(b,a)i; a+bi−−−−−√=a2+b2−−−−−−√+a2−−−−−−−−−−−√+i⋅sgn(b)a2+b2−−−−−−√−a2−−−−−−−−−−−√;
For general complex numbers with a,b≠0:
ax=exlna, 1a=e−lna; a0=1, a−x=1ax; axay=ax+y, axay=ax−y;
*(ax)y=axy(a>0,x∈R);
*(ab)x=axbx,ab−−√=a−−√b√(R(a),R(b)≥0, a,b not both negative imaginary;a,b∈R, a,b not both negative);
*(ab)x=axbx,ab−−√=a−−√b√(R(a),R(b)>0; a,b∈R), *1a−−√=1a−−√(a not negative real);
a2−−√={a,−a,R(a)>0; R(a)=0,I(a)>0R(a)<0; R(a)=0,I(a)<0; (a−−√)2=a;
eln(x)=x; ln(ex)=x(I(x)<π); aln(x)/ln(a)=x(x≠0,a≠0,1);
*ln(xy)=ln(x)+ln(y)(R(x),R(y)>0;x,y∈R, x,y not both negative), R(ln(xy))=R(ln(x)+ln(y))(x,y≠0);
*ln(xy)=ln(x)−ln(y)(R(x),R(y)>0;x,y∈R), R(ln(xy))=R(ln(x)−ln(y))(x,y≠0); *ln(1x)=−ln(x);
**ln(xa)=aln(x)(a,x∈R,x>0); *lnx−−√=12ln(x)(x not negative real);
Properties of complex conjugate
z±w¯¯¯¯¯¯¯¯¯¯¯¯=z¯¯¯±w¯¯¯¯,zw¯¯¯¯¯¯=z¯¯¯⋅w¯¯¯¯,(zw)¯¯¯¯¯¯¯¯¯¯¯=z¯¯¯w¯¯¯¯;zz¯¯¯=|z|2,z−1=z¯¯¯|z|2,zn¯¯¯¯¯=(z¯¯¯)n (n∈Z),exp(z¯¯¯)=exp(z)¯¯¯¯¯¯¯¯¯¯¯¯¯¯,ln(z¯¯¯)=ln(z)¯¯¯¯¯¯¯¯¯¯;
Hyperbolic Functions
sinhx=ex−e−x2=e2x−12ex=1−e−2x2e−x,coshx=ex+e−x2=e2x+12ex=1+e−2x2e−x;
tanhx=sinhxcoshx=ex−e−xex+e−x=e2x−1e2x+1=1−2e2x+1,cothx=coshxsinhx=ex+e−xex−e−x=e2x+1e2x−1=1+2e2x−1;
sinh−1(x)=ln(x+x2+1−−−−−√),cosh−1(x)=ln(x±x2−1−−−−−√),tanh−1(x)=12ln(1+x1−x),coth−1(x)=12ln(x+1x−1);
f(x)=sinhx
f(x)=coshx
f(x)=tanhx
f(x)=cschx
f(x)=sechx
f(x)=cothxsin(ix)=i⋅sinhx,cos(ix)=coshx,tan(ix)=i⋅tanhx;sinh(ix)=i⋅sinx,cosh(ix)=cosx,tanh(ix)=i⋅tanx;
sin−1(iz)=i⋅sinh−1(z),tan−1(iz)=i⋅tanh−1(z);sinh−1(iz)=i⋅sin−1(z),tanh−1(iz)=i⋅tan−1(z);
coshx+sinhx=ex,coshx−sinhx=e−x;cosh2x−sinh2x=1,sech2x+tanh2x=1,coth2x−csch2x=1;
sinh(a±b)=sinhacoshb±coshasinhb,cosh(a±b)=coshacoshb±sinhasinhb,tanh(a±b)=tanha±tanhb1±tanhatanhb;
sinh2x=2sinhxcoshx,cosh2x=sinh2x+cosh2x=2sinh2x+1=2cosh2x−1,tanh2x=2tanhx1+tanh2x;
sinh3x=⎧⎩⎨⎪⎪4sinh3x+3sinhx4sinhxcosh2x−sinhx3sinhxcosh2x+sinh3x,cosh3x=⎧⎩⎨⎪⎪4cosh3x−3coshx4sinh2xcoshx+coshx3sinh2xcoshx+cosh3x,tanh3x=3tanhx+tanh3x1+3tanh2x;
sinhx⋅sinhy=12(cosh(x+y)−cosh(x−y)),coshx⋅coshy=12(cosh(x+y)+cosh(x−y)),sinhx⋅coshy=12(sinh(x+y)+sinh(x−y));
sinh2(x)=12(cosh(2x)−1),cosh2(x)=12(cosh(2x)+1),sinh(x)⋅cosh(x)=12sinh(2x);
sinhx±sinhy=2sinh(x±y2)cosh(x∓y2);coshx+coshy=2cosh(x+y2)cosh(x−y2),coshx−coshy=2sinh(x+y2)sinh(x−y2);
sinh(cosh−1x)=x2−1−−−−−√,cosh(sinh−1x)=x2+1−−−−−√,tanh(sinh−1x)=xx2+1−−−−−√,tanh(cosh−1x)=x2−1−−−−−√x,sinh(tanh−1x)=x1−x2−−−−−√,cosh(tanh−1x)=11−x2−−−−−√;(x∈R);
Limit
lim[f(x)±g(x)]=limf(x)±limg(x);lim[f(x)⋅g(x)]=limf(x)⋅limg(x);limf(x)g(x)=limf(x)limg(x);lim[c⋅f(x)]=c⋅limf(x);lim[f(x)]n=[limf(x)]n,n∈N;
If yn≤xn≤zn and limyn=limzn=a, then limxn=a;
If limf(x)g(x)=1, then lim[f(x)⋅h(x)]=lim[g(x)⋅h(x)], limh(x)f(x)=limh(x)g(x);
L'Hôpital's Rule: If limf(x)=limg(x)=0 or limf(x)=limg(x)=∞, then limf(x)g(x)=limf′(x)g′(x);
L'Hôpital's Rule for other indeterminates: lim0⋅∞⇒lim01/∞,lim∞1−∞2⇒lim1/∞1−1/∞21/(∞1∞2),lim∞0⇒exp(lim01/ln∞),lim1∞⇒exp(limln11/∞),lim(0+)0⇒exp(lim01/ln0+),limlog00⇒limln0ln0,limlog11⇒limln1ln1;
Derivative
ddx[a⋅f(x)+b⋅g(x)]=a⋅f′(x)+b⋅g′(x);ddx[f(x)⋅g(x)]=f′(x)⋅g(x)+f(x)⋅g′(x);
ddx[f1(x)⋅f2(x)⋅f3(x)]=f′1(x)⋅f2(x)⋅f3(x)+f1(x)⋅f′2(x)⋅f3(x)+f1(x)⋅f2(x)⋅f′3(x); “Each item take turns at derivation”
ddx(f(x)g(x))=f′(x)g(x)−f(x)g′(x)g2(x),ddx(1u(x))=u′(x)u2(x);
ddx(f∘g)(x)=(f′∘g)(x)⋅g′(x);ddxf−1(x)=1(f′∘f−1)(x);dndnx[f(x)⋅g(x)]=∑k=0n(nk)⋅f(n−k)(x)⋅g(k)(x);
ddxC=0;ddx|x|=x|x|;
ddxxμ=μxμ−1;ddxex=ex,ddxax=axlna;ddx1x=−1x2,ddxx−−√=12x−−√,ddxx−−√3=13x−23;
ddxln|x|=1x,ddxloga|x|=1xlna;
ddxsinx=cosx,ddxcosx=−sinx,ddxtanx=sec2x,ddxcscx=−cscxcotx,ddxsecx=secxtanx,ddxcotx=−csc2x;
ddxsin−1x=11−x2−−−−−√,ddxcos−1x=−11−x2−−−−−√,ddxtan−1x=11+x2,ddxcot−1x=−11+x2;
ddxsinhx=coshx,ddxcoshx=sinhx,ddxtanhx=sech2x,ddxcschx=−cothx⋅cschx,ddxsechx=−sechx⋅tanhx,ddxcothx=−csch2x;
ddxsinh−1x=1x2+1−−−−−√,ddxcosh−1x=1x2−1−−−−−√,ddxtanh−1x=ddxcoth−1x=11−x2;
ddxerf(x)=2π−−√e−x2;
Incomplete Mark
Taylor Series: f(x)=∑k=0n(x−x0)k⋅f(k)(x0)k!+Rn; Maclaurin Series: f(x)=∑k=0nxk⋅f(k)(0)k!+Rn;
Arc length and curvature
(x(t),y(t)): ds=x′2+y′2−−−−−−−√dt; (x(t),y(t),z(t)): ds=x′2+y′2+z′2−−−−−−−−−−−√dt; y=f(x): ds=y′2+1−−−−−√dx; r=r(θ): ds=r2+r′2−−−−−−√dθ;
Curvature: κ=limΔs→0ΔθΔs=∣∣∣dθds∣∣∣;
r⃗ =r(t): κ(t)=|r′(t)×r′′(t)||r′(t)|3; y=f(x): κ(x)=y′′(1+y′2)32; r=r(θ): κ(θ)=r2+2r′2−rr′′(r2+r′2)32;
Unit tangent: T=r′(t)|r′(t)|; Normal unit vector: N=T′(t)|T′(t)|; Binormal unit vector: B=T×N;
Frenet–Serret formula: ⎡⎣⎢T′N′B′⎤⎦⎥=⎡⎣⎢0−κ0κ0−τ0τ0⎤⎦⎥⎡⎣⎢TNB⎤⎦⎥, where τ=−dBds⋅N is the torsion;
Kinematics
Velocity and acceleration in Cartesian coordinate: p⃗ =xi⃗ +yj⃗ +zk⃗ , v⃗ =dxdti⃗ +dydtj⃗ +dzdtk⃗ , a⃗ =d2xdt2i⃗ +d2ydt2j⃗ +d2zdt2k⃗ ;
Velocity and acceleration in polar coordinate {r=r(t)θ=θ(t): p⃗ =rer→, v⃗ =r˙er→+rθ˙eθ→, a⃗ =(r¨−rθ˙2)er→+(2r˙θ˙+rθ¨)eθ→;
Velocity and acceleration in eigen coordinate: v⃗ =vet→, a⃗ =dvdtet→+v2Ren→, R is the curvature radius;
(a⃗ can also be calculated by projecting acceleration vector in et→ and en→ directions)
Derivatives of unit vectors in polar coordinate: ddter→=θ˙eθ→, ddteθ→=−θ˙er→;
Integral
∫a⋅f(x)±b⋅g(x)dx=a∫f(x)±b∫g(x)dx;
∫(f∘g)(x)⋅g′(x)dx=∫f(u)du|u=g(x),∫(f∘g)(x)dx=∫1g′(x)(f∘g)(x)d[g(x)];
∫f(x)dx=∫(f∘g)(u)⋅g′(u)du|u=g−1(x);
∫uv′=uv−∫u′v; （Log / Inverse trig − Algebraic − Trigonometric − Exp）
∫kdx=kx+C;∫|x|dx=x|x|2+C;∫1xdx=ln|x|+C,∫xμdx=xμ+1μ+1+C(μ≠1);∫exdx=ex+C,∫axdx=axlna+C;∫x−−√dx=23x3/2+C;
∫sinxdx=−cosx+C,∫cosxdx=sinx+C,∫tanxdx=−ln|cosx|+C;∫cotxdx=ln|sinx|+C,∫secxdx=tanh−1(sinx)+C,∫cscxdx=ln∣∣tanx2∣∣+C=−tanh−1(cosx)+C;
∫sinhxdx=coshx+C,∫coshxdx=sinhx+C,∫tanhxdx=−ln(coshx)+C;∫cothxdx=ln|sinhx|+C,∫sechxdx=tan−1(sinhx)+C,∫cschxdx=−coth−1(coshx)+C=ln(|e−x−1|)−ln(|e−x+1|)+C;
∫lnxdx=x(ln(x)−1)+C;∫sin−1(x)dx=xsin−1(x)+1−x2−−−−−√+C,∫cos−1(x)dx=xcos−1(x)−1−x2−−−−−√+C;
∫tan−1(x)dx=xtan−1(x)−ln(x2+1)2+C,∫cot−1(x)dx=xcot−1(x)+ln(x2+1)2+C; ∫e−x2dx=π−−√2erf(x)+C;
∫1a2+x2dx=1atan−1(xa)+C,∫1a2−x2dx=12aln∣∣∣x+ax−a∣∣∣+C=1atanh−1(xa)+C=1acoth−1(xa)+C;
∫1a2+x2−−−−−−√dx=sinh−1(x|a|)+C,∫1a2−x2−−−−−−√dx=sin−1(x|a|)+C,∫1x2±a2−−−−−−√dx=ln∣∣x2±a2−−−−−−√+x∣∣+C;
∫a2−x2−−−−−−√dx=12(a2sin−1(x|a|)+xa2−x2−−−−−−√)+C;∫a2+x2−−−−−−√dx=12(a2sinh−1(x|a|)+xa2+x2−−−−−−√)+C;
∫x2−a2−−−−−−√dx=12(xx2−a2−−−−−−√−a2cosh−1(x|a|))+C;(x>0)
∫xexdx=xex−ex+C;∫xsin(x)dx=sin(x)−xcos(x)+C,∫xcos(x)dx=cos(x)+xsin(x)+C;
Incomplete Mark
∫aaf(x)dx=0; ∫baf(x)dx=−∫abf(x)dx; ∫caf(x)dx=∫baf(x)dx+∫cbf(x)dx;
∫ba[f(x)±g(x)]dx=∫baf(x)dx±∫bag(x)dx, ∫bak⋅f(x)dx=k⋅∫baf(x)dx, ∫bak1f1(x)±k2f2(x)dx=k1∫baf1(x)dx±k2∫baf2(x)dx;
ddx(∫ψ(x)φ(x)f(t)dt)=ψ′(x)f[ψ(x)]−φ′(x)f[φ(x)];
∫baf(x)dx=∫φ−1(b)φ−1(a)f[φ(t)]φ′(t)dt, ∫baf[φ(x)]φ′(x)dx=∫φ(b)φ(a)f(x)dx=∫baf[φ(x)]dφ(x); ∫π20f(sinx)dx=∫π20f(cosx)dx;
Newton-Leibniz Formula: ∫baf(x)dx=F(b)−F(a);
Green Formula: ∮LPdx+Qdy=∬D(∂Q∂x−∂P∂y)dxdy; (A=12∮Lxdy−ydx);
Gauss Formula: ⬭∬Σ+Pdydz+Qdxdz+Rdxdy=∭V(∂P∂x+∂Q∂y+∂R∂z)dxdydz; (V=13⬭∬Σxdydz+ydzdx+zdxdy);
Stokes Formula: ∮LPdx+Qdy+Rdz=∬Σ(∂R∂y−∂Q∂z)dydz+(∂P∂z−∂R∂x)dzdx+(∂Q∂x−∂P∂y)dxdy;
Special Integrals
Elliptic integral of the first kind: F(φ,k)=F(φ|k2)=F(sinφ;k)=∫φ0dθ1−k2sin2θ−−−−−−−−−−√; K(k)=∫π20dθ1−k2sin2θ−−−−−−−−−−√=∫10dt(1−t2)(1−k2t2)−−−−−−−−−−−−−−√;
Elliptic integral of the second kind: E(φ,k)=E(φ|k2)=E(sinφ;k)=∫φ01−k2sin2θ−−−−−−−−−−√dθ; E(k)=∫π201−k2sin2θ−−−−−−−−−−√dθ=∫101−k2t2−−−−−−−√1−t2−−−−−√dt;
Euler's Integrals: B(x,y)=∫10tx−1(1−t)y−1dt, Γ(x)=∫∞0tx−1e−tdt; B(x,y)=B(y,x)=Γ(x)Γ(y)Γ(x+y);
Γ(x+1)=xΓ(x); B(x+1,y+1)=xy(x+y)(x+y+1)B(x,y), B(x+1,y)=xx+yB(x,y); Γ(x)Γ(1−x)=πsin(πx), Γ(2x)=22x−1π−−√Γ(x)Γ(x+12);
∫π20cosm(x)sinn(x)dx=12B(m+12,n+12)=Γ(m+12)Γ(n+12)2⋅Γ(m+n2+1), ∫π20sinn(x)dx=∫π20cosn(x)dx=⎧⎩⎨⎪⎪⎪⎪(n−1)!!n!!π2,(n−1)!!n!!,nevennodd, ∫π20tanα(x)dx=π2cos(απ2);
Incomplete Mark
Multivariable Derivative
∂[f1f2⋯]T∂[x1x2⋯]T=⎡⎣⎢⎢⎢⎢⎢∂f1∂x1∂f2∂x1⋮∂f1∂x2∂f2∂x2⋮⋯⋯⋱⎤⎦⎥⎥⎥⎥⎥; ∂(u,v)∂(x,y)=∣∣∣∣∂u∂x∂v∂x∂u∂y∂v∂y∣∣∣∣, ∂(u,v,w)∂(x,y,z)=∣∣∣∣∣∣∂u∂x∂v∂x∂w∂x∂u∂y∂v∂y∂w∂y∂u∂z∂v∂z∂w∂z∣∣∣∣∣∣;
∂(u,v)∂(s,t)=∂(u,v)∂(x,y)⋅∂(x,y)∂(s,t), ∂(u,v)∂(x,y)⋅∂(x,y)∂(u,v)=1, ∂(u,v)∂(x,y)=−∂(v,u)∂(x,y), ∂(u,u)∂(x,y)=0;
Area and volume elements ⭧
dA=dxdy=∂(x,y)∂(u,v)dudv, dV=dxdydz=∂(x,y,z)∂(u,v,w)dudvdw;
Polar/cylindrical xy=ρcos(θ)=ρsin(θ): ∂(x,y)∂(ρ,θ)=ρ; Spherical xyz=ρsin(φ)cos(θ)=ρsin(φ)sin(θ)=ρcos(φ): ∂(x,y,z)∂(ρ,θ,φ)=ρ2sin(φ);
Parametric surface r=r(u,v): dA=|r′u×r′v|dudv=r′2ur′2v−(r′u⋅r′v)2−−−−−−−−−−−−−√ dudv;
Derivative of implicit functions
F(x,y(x))=0: dydx=−∂F∂x∂F∂y=−F′x/F′y, d2ydx2=−∂2F∂x2(∂F∂y)2−2∂2F∂x∂y∂F∂x∂F∂y+∂2F∂y2(∂F∂x)2(∂F∂y)3;
F(x,y,z(x,y))=0: dzdx=−F′x/F′z, dzdy=−F′y/F′z;
Vector Calculus
Gradient of a scalar field: grad u=∇u=∂u∂xi+∂u∂yj+∂u∂zk; Directional derivative: ddtu(x0+l⋅t)=grad u(x0)⋅l;
∇(C1u1+C2u2)=C1∇u1+C2∇u2, ∇u1u2=u1∇u2+u2∇u1, ∇f(u)=f′(u)∇u;
Divergence of a vector field: div F=∇⋅F=∂Fx∂x+∂Fy∂y+∂Fz∂z;
div(C1v1+C2v2)=C1div(v1)+C2div(v2), div(uv)=u⋅div(v)+v⋅grad(u);
Curl of a vector field: curl F=∇×F=∣∣∣∣∣i∂∂xFxj∂∂yFyk∂∂zFz∣∣∣∣∣=(∂Fz∂y−∂Fy∂z)i+(∂Fx∂z−∂Fz∂x)j+(∂Fy∂x−∂Fx∂y)k;
Incomplete Mark
⭧*∇|r|=r|r|; ∇⋅r|r|=∑ir⋅r−r2i|r|3=dim(r)−1|r|,
Incomplete Mark
Differential equations
Solve dydx=f(x): y=∫f(x)dx+C;
Solve dydx=f(x)g(y): 1g(y)dy=f(x)dx⟹∫1g(y)dy=∫f(x)dx+C;
Solve dydx=φ(yx): z=yx⟹xdzdx+z=φ(z)⟹dzdx=φ(z)−zx; φ(z)−z≠0:∫dzφ(z)−z=ln∣∣xC∣∣;φ(z)−z≡0:z=yx=C;
Solve dydx+p(x)y=q(x): y(x)=e−∫p(x)dx[∫q(x)e∫p(x)dxdx+C];
Solve dydx+p(x)y=q(x)yn: z=y1−n⟹dzdx+(1−n)p(x)z=(1−n)q(x); (treat as above)
Solve y(n)=f(x): dn−1ydxn−1=∫f(x)dx+C1; (repeat)
Solve y′′=f(x,y′): p(x)=dydx⟹dpdx=f(x,p); p(x)=φ(x,C1)⟹y=∫φ(x,C1)dx+C2;
Solve y′′=f(y,y′): p(y)=dydx⟹d2ydx2=pdpdy=f(y,p); p(y)=φ(y,C1)⟹∫dyφ(y,C1)=x+C2;
Solve y(n)=f(x,y(n−1)): u(x)=y(n−2)⟹u′′=f(x,u′), solve using above and repeat;
Solve y(n)=f(y(n−1),y(n−2)): u(x)=y(n−2)⟹u′′=f(u′,u), solve using above and repeat;
Solve y′′+py′+qy=0: Let r be the roots of r2+pr+q=0; y=⎧⎩⎨C1er1x+C2er2x,erx(C1+C2x),eax(C1cos(bx)+C2sin(bx)),r∈[r1,r2], r1≠r2r=r1=r2r=a±bi, b≠0
Solve y′′+py′+qy=eλxP(x): Find Q(x) such that Q′′(x)+(2λ+p)Q′(x)+(λ2+pλ+q)Q(x)=P(x), ⟹y=eλxQ(x)+(solution of y′′+py′+qy=0);
λ2+pλ+q and 2λ+p are both zero when λ is a multiple root of λ2+pλ+q=0;
P(x)=Pm(x) is a degree-m polynomial: Q(x)=⎧⎩⎨⎪⎪Qm(x),xQm(x),x2Qm(x),λ2+pλ+q≠0λ2+pλ+q=0, 2λ+p≠0λ2+pλ+q=2λ+p=0, where Qm(x) is a polynomial of degree m;
eλxP(x)=eax[Pm(x)cos(bx)+Pn(x)sin(bx)]: eλxQ(x)=xkeax[R1(x)cos(bx)+R2(x)sin(bx)], k={0,1,(a+bi)2+p(a+bi)+q≠0(a+bi)2+p(a+bi)+q=0, Ri(x) has degree max(m,n);
Solve y′′+p(x)y′+q(x)y=0: Find any non-zero solution y1(x), let y2(x)=y1(x)∫e−∫p(x)dxy1(x)2dx, ⟹y=C1y1(x)+C2y2(x);
Solve y′′+p(x)y′+q(x)y=f(x): Find any solution y∗(x), ⟹y(x)=y∗(x)+(solution of y′′+p(x)y′+q(x)y=0);
If y=y1(x) satisfies y′′+p(x)y′+q(x)y=f1(x) and y=y2(x) satisfies y′′+p(x)y′+q(x)y=f2(x),
then y=C1y1(x)+C2y2(x) satisfies y′′+p(x)y′+q(x)y=C1f1(x)+C2f2(x);
Solve x2y′′+pxy′+qy=f(x): x=et⟹d2ydt2+(p−1)dydt+qy=f(et);
Incomplete Mark
Matrix
(AB)C=A(BC), k(AB)=(kA)B=A(kB), A(B+C)=AB+AC, (B+C)A=BA+CA; AI=IA=A; AmAn=Am+n, (Am)n=Amn;
(AT)T=A, (A+B)T=AT+BT, (kA)T=kAT, (AB)T=BTAT, (A1A2…Ak)T=ATkATk−1…AT1; ATA=0⇒A=O;
A invertible: ⟹AB=O⇒A=O, AX=AY⇒X=Y, AX=O⇒X=O, AX=B⇒X=A−1B;
A,B invertible: ⟹(A−1)−1=A, (λA)−1=λ−1A−1, (AB)−1=B−1A−1, (AT)−1=(A−1)T;
detAB=detA⋅detB, detAT=detA, detA−1=1detA;
Swap two rows/cols: detA′=−detA; Multiply a row/col by k: detA′=kdetA; Add the multiple of a row/col to another row/col: detA′=detA;
det⎛⎝⎜⎜a1O⋱∗an⎞⎠⎟⎟=a1a2…an, det⎛⎝⎜⎜A1O⋱∗An⎞⎠⎟⎟=|A1||A2|…|An|, detI=1;
∣∣∣∣∣⋯x1a+x1b⋯⋯⋯⋯⋯xna+xnb⋯∣∣∣∣∣=∣∣∣∣∣⋯x1a⋯⋯⋯⋯⋯xna⋯∣∣∣∣∣+∣∣∣∣∣⋯x1b⋯⋯⋯⋯⋯xnb⋯∣∣∣∣∣; Vandermonde Determinant: ∣∣∣∣∣∣∣∣1x1x21⋮xn−111x2x22⋮xn−121x3x23⋮xn−13⋯⋯⋯⋱⋯1xnx2n⋮xn−1n∣∣∣∣∣∣∣∣=∏i<j(xj−xi);
A−1=A∗|A|; AA∗=A∗A=det(A)⋅I; A−1=[acbd]−1=1|A|[d−c−ba];
Cramer's Rule: Solution of AX=⎡⎣⎢⎢⎢⎢⎢⎢a11a21⋮an1a12a22⋮an2⋯⋯⋱⋯a1na2n⋮ann⎤⎦⎥⎥⎥⎥⎥⎥⎡⎣⎢⎢⎢⎢⎢⎢x1x2⋮xn⎤⎦⎥⎥⎥⎥⎥⎥=⎡⎣⎢⎢⎢⎢⎢⎢y1y2⋮yn⎤⎦⎥⎥⎥⎥⎥⎥=Y is xi=|Ai||A|, which Ai is the matrix formed by replacing the ith column of A by Y;
R(A)=0⟺A=O, R(An×n)=n⟺Aintertible; R(kA)=R(A)(k≠0), R(AT)=R(A), R(A∗)=⎧⎩⎨n,1,0,R(A)=nR(A)=n−1R(A)<n−1; R[AOOB]=R(A)+R(B);
Elementary transforms don’t change rank of a matrix; Am×n≅[IR(A)OOO]m×n;
Incomplete Mark
2×2 matrix
Let A=[a11a21a12a22]. detA=a11a22−a12a21, A−1=1detA[a22−a21−a12a11]; R(A)=1⟺A rows and columns in proportion, R(A)=0⟺A=O.
Characteristic polynomial: λ2−(a11+a22)λ+(a11a22−a12a21); λI−A=[λ−a11−a21−a12λ−a22]
Eigenvalues: λ=12((a11+a22)±(a11−a22)2+4a12a21−−−−−−−−−−−−−−−−−√);
* Eigenvectors: (λ−a22,a21)∥((a11−a22)±(a11−a22)2+4a12a21−−−−−−−−−−−−−−−−−√, 2a21), (a12,λ−a11)∥(2a12, (−a11+a22)±(a11−a22)2+4a12a21−−−−−−−−−−−−−−−−−√);
*An=λn1−λn2λ1−λ2A−λ1λ2λn−11−λn−12λ1−λ2I;
Vector
a⃗ +b⃗ =b⃗ +a⃗ , (a⃗ +b⃗ )+c⃗ =a⃗ +(b⃗ +c⃗ ), k(l⋅a⃗ )=(kl)⋅a⃗ , k(a⃗ +b⃗ )=ka⃗ +kb⃗ ;
|a⃗ |=a21+a22+a23−−−−−−−−−−√, |k⋅a⃗ |=|k|⋅|a⃗ |; |Prjua⃗ |=|a⃗ |⋅cos⟨a⃗ ,u⟩, Prju(a⃗ 1+a⃗ 2)=Prjua⃗ 1+Prjua⃗ 2; |a⃗ ±b⃗ |≤|a⃗ |+|b⃗ |; |a⃗ ⋅b⃗ |≤|a⃗ |⋅|b⃗ |;
cosθx=a1|a⃗ |, cosθy=a2|a⃗ |, cosθz=a3|a⃗ |; e⃗ a⃗ =a⃗ |a⃗ |=(cosθx,cosθy,cosθz); cos2θx+cos2θy+cos2θz=1;
a⃗ ⋅b⃗ =|a⃗ |⋅|b⃗ |⋅cosθ=a1b1+a2b2+a3b3, a⃗ ⋅b⃗ =b⃗ ⋅a⃗ , (λ⋅a⃗ )⋅b⃗ =a⃗ ⋅(λ⋅b⃗ )=λ⋅(a⃗ ⋅b⃗ ), (a⃗ ±b⃗ )⋅c⃗ =a⃗ ⋅b⃗ ±a⃗ ⋅c⃗ ; (a⃗ ±b⃗ )2=a⃗ 2±2a⃗ ⋅b⃗ +b⃗ 2, a⃗ 2=|a⃗ |2;
a⃗ ⊥b⃗ ⟺a⃗ ⋅b⃗ =0⟺a1b1+a2b2+a3b3=0; ∣∣Prja⃗ b⃗ ∣∣=a⃗ ⋅b⃗ |a⃗ |=b⃗ ⋅e⃗ a⃗ ; ⟨a⃗ ,b⃗ ⟩=cos−1a⃗ ⋅b⃗ |a⃗ |⋅|b⃗ |=cos−1a1b1+a2b2+a3b3a21+a22+a23−−−−−−−−−−√b21+b22+b33−−−−−−−−−√, a⃗ ⋅b⃗ ⎧⎩⎨⎪⎪>0,<0,⟨a⃗ ,b⃗ ⟩<π2⟨a⃗ ,b⃗ ⟩>π2;
a⃗ ×b⃗ =c⃗ {c⃗ ⊥a⃗ ,c⃗ ⊥b⃗ ,right-hand rule|c⃗ |=|a⃗ |⋅|b⃗ |⋅sinθ=∣∣∣∣∣i⃗ a1b1j⃗ a2b1k⃗ a3b3∣∣∣∣∣; a⃗ ∥b⃗ ⟺a⃗ ×b⃗ =0, a⃗ ⊥b⃗ ⟺|a⃗ ×b⃗ |=|a⃗ ||b⃗ |;
a⃗ ×b⃗ =−b⃗ ×a⃗ , a⃗ ×a⃗ =0; (λa⃗ )×b⃗ =λ(a⃗ ×b⃗ ); (a⃗ ±b⃗ )×c⃗ =a⃗ ×c⃗ ±b⃗ ×c⃗ , c⃗ ×(a⃗ ±b⃗ )=c⃗ ×a⃗ ±c⃗ ×b⃗ ;
[a⃗ ,b⃗ ,c⃗ ]=(a⃗ ×b⃗ )⋅c⃗ =∣∣∣∣a1b1c1a2b2c2a3b3c3∣∣∣∣; [a⃗ ,b⃗ ,c⃗ ]=[b⃗ ,c⃗ ,a⃗ ]=[c⃗ ,a⃗ ,b⃗ ]; [a⃗ ,b⃗ ,(λc⃗ +μd⃗ )]=λ[a⃗ ,b⃗ ,c⃗ ]+μ[a⃗ ,b⃗ ,d⃗ ];
Analytic Geometry
Basic Equations
Scylinder=2πr2+2πrl, Scone=πr2+πrl, Ssphere=4πr2; Vcylinder=Sh, Vcone=13Sh, Vsphere=43πR3;
(ρ;θ)⟹(ρ⋅cosθ,ρ⋅sinθ), (x,y)⟹(x2+y2−−−−−−√;atan2(y,x));
In any triangle: asinA=bsinB=csinC=2R; c2=a2+b2−2abcosC, cosA=b2+c2−a22bc; where R=abc4⋅Area is the circumradius of the triangle;
Area=p(p−a)(p−b)(p−c)−−−−−−−−−−−−−−−−−√=rs, where s=12(a+b+c), r denotes the inradius;
Area of quadrilateral: (s−a)(s−b)(s−c)(s−d)−abcdcos2θ−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√, where s=14(a+b+c+d), cosθ=cos[12(∠A+∠C)]=cos[12(∠B+∠D)];
Distance from point (x0,y0) to line Ax+By+C=0: d=|Ax0+By0+C|A2+B2−−−−−−−√;
Distance from point (x0,y0,z0) to plane Ax+By+Cz+D=0: d=|Ax0+By0+Cz0+D|A2+B2+C2−−−−−−−−−−−√;
Distance between y=mx+b and y=mx+c: |b−c|m2+1−−−−−−√;
Equations of Planar Straight Line
| 0) | General | Ax+By+C=0 | m=−AB, a=−CA, b=−CB, n⃗ =(A,B) |
| 1) | Slope m and y-intercept b | y=mx+b | a=−bm |
| 2) | Point (x0,y0) and slope m | y−y0=m(x−x0) | a=−y0m+x0, b=y0−mx0 |
| 3) | Points (x1,y1) and (x2,y2) | y−y1y2−y1=x−x1x2−x1 | m=y2−y1x2−x1, a=x1y2−x2y1y2−y1, b=x2y1−x1y2x2−x1 |
| 4) | x and y intercepts a, b | xa+yb=1 | m=−ba |
m is the slope and a, b are x and y intercepts. l1∥l2⟺m1=m2, l1=l2⟺m1=m2∧b1=b2; l1⊥l2⟺m1m2=−1; |
Equations of Circle
| 1) | Center C(a,b) and radius r | (x−a)2+(y−b)2=r2 | ⇒x2+y2+(−2a)x+(−2b)y+(a2+b2−r2)=0 |
| 2) | General | x2+y2+Dx+Ey+F=0
(D2+E2−4F>0) | C=(−D2,−E2), r=12D2+E2−4F−−−−−−−−−−−−√ |
Equation 2) is convenient in determining the equation of a circle through 3 given points. (Solve systems of linear equations. )
Quadratic Curves
Ellipse: x2a2+y2b2=1(a>b>0); x∈[−a,a], y∈[−b,b], c=a2−b2−−−−−−√, F=(±c,0), e=ca<1;
Hyperbola: x2a2−y2b2=1(a,b>0); |x|≥a, c=a2+b2−−−−−−√, F=(±c,0), e=ca>1; asymptote: xa±yb=0;
Parabola: y2=2px(p>0); Focus: (p2,0), Directrix: x=−p2, e=1; y=ax2: F=14a, directrix: y=−14a;
Equations of Plane
| 0) | General | Ax+By+Cz+D=0 | n⃗ =(A,B,C) |
| 1) | Normal n⃗ =(A,B,C)
and point P=(x0,y0,z0) | A(x−x0)+B(y−y0)+C(z−z0)=0
⟹Ax+By+Cz=Ax0+By0+Cz0 | |
| 2) | Three points Pi(xi,yi,zi)
(i={1,2,3}) | ∣∣∣∣x−x1x2−x1x3−x1y−y1y2−y1y3−y1z−z1z2−z1z3−z1∣∣∣∣=0 | |
| 3) | x, y and z intercepts a, b, c | xa+yb+zc=1 | |
A=0: parallel to x-axis; A=B=0: parallel to xOy plane; D=0: through Origin; π1⊥π2⟺A1A2+B1B2+C1C2=0, π1∥π2⟺A1A2=B1B2=C1C2; ⟨π1,π2⟩=cos−1|A1A2+B1B2+C1C2|A21+B21+C21−−−−−−−−−−−√⋅A22+B22+C22−−−−−−−−−−−√; |
Equations of Spatial Straight Line
| 1) | Point P(x0,y0,z0)
and direction vector s⃗ =(a,b,c) | x−x0a=y−y0b=z−z0c | If a,b,or c occurs as 0, then the numerator is also 0 |
| 2) | through P1(x1,y1,z1)
and P2(x2,y2,z2) | x−x1x2−x1=y−y1y2−y1=z−z1z2−z1 | |
| 3) | Parameter | ⎧⎩⎨x=x0+a⋅ty=y0+b⋅tz=z0+c⋅t | P0(x0,y0,z0) and direction vector s⃗ =(a,b,c), same as #1;
Usually for solving equations |
| 4) | Intersection of two planes | {A1x+B1y+C1z+D1=0A2x+B2y+C2z+D2=0 | s⃗ =∣∣∣∣∣i⃗ A1A2j⃗ B1B2k⃗ C1C2∣∣∣∣∣; |
Distance from point M to line P=P0+s⃗ ⋅t: d=|s⃗ ×MP0→||s⃗ |;
l1∥l2⟺s⃗ 1∥s⃗ 2, l1=l2⟺s⃗ 1∥s⃗ 2∥P1P2→; l1 intersect l2⟺s⃗ 1∦s⃗ 2∧[s1→,s2→,P1P2→]=0; ⟨l1,l2⟩=cos−1|s⃗ 1⋅s⃗ 2||s⃗ 1|⋅|s⃗ 2|, l1⊥l2⟺s⃗ 1⋅s⃗ 2=0;
l∥π⟺s⃗ ⋅n⃗ =0, l⊂π⟺s⃗ ⋅n⃗ =0∧Ax0+By0+Cz0+D=0; l intersect π⟺s⃗ ⋅n⃗ ≠0, ⟨l,π⟩=sin−1|s⃗ ⋅n⃗ ||s⃗ |⋅|n⃗ |;
Sheaf of Planes: Let the equation of straight line l be {A1x+B1y+C1z+D1=0(1)A2x+B2y+C2z+D2=0(2),
the equation of all planes through l is given by: (A1x+B1y+C1z+D1)+λ(A2x+B2y+C2z+D2)=0, except #2 (λ→∞);
Incomplete Mark
Permutation and Combination
n different types, k choices:
Number of Orders: n!;
Permutations without Repetition: Pnk=n!(n−k)!; (Also denoted as Pkn, Akn, nPk, nPk)
Permutations with Repetition: Unk=nk;
Combinations without Repetition: Cnk=n!k!(n−k)!; (Also denoted as C(n,k), (nk), Ckn, nCk, nCk)
Combinations with Repetition: Hnk=(n+k−1)!k!(n−1)!; (Also denoted as Fnk) Hnk=Cn+k−1k=Cn+k−1n−1;
Number of Orders in Circular Permutation: (n−1)!;
*Circular Permutation without Repetition: Qnk=n!k⋅(n−k)!; Qnk=Pnkk;
*Circular Permutation with Repetition: ∑r|k(r⋅φ(r)⋅nkr)k;
(nk)=(nn−k)=nk(n−1k−1);(nk−1)+(nk)=(n+1k),(nk)=(n−1k−1)+(n−1k);
Statistics and Probability
Mean: μ=1N∑i=1Nxi; Variance: σ2=1N∑i=0N(xi−μ)2=1N⎛⎝∑i=1Nx2i−1N(∑i=1Nxi)2⎞⎠; Standard deviation: σ=σ2−−√;
Variance of a sample: s2=1n−1∑i=0n(xi−μ)2=1n−1⎛⎝∑i=1nx2i−1n(∑i=1nxi)2⎞⎠; Standard deviation of a sample: s=s2−−√;
Random variable, expected value and variance
Discrete random variable: E(X)=∑ixiP(xi),var(X)=∑i(xi−μX)P(xi);
Continuous random variable: E(X)=∫∞−∞x⋅pX(x)dx,var(X)=∫∞−∞(x−μX)2⋅pX(x)dx;
E(aX+b)=aE(X)+b,var(aX+b)=a2var(X),σaX+b=|a|⋅σX; var(X)=E(X2)−E(X)2; E(X±Y)=E(X)±E(Y);
For independent random variables X, Y: E(XY)=E(X)E(Y);var(X+Y)=var(X)+var(Y), σX+Y=σ2X+σ2Y−−−−−−−√;
*Probability distributions
Normal distribution: PDF(x)=1σ2π−−√exp(−(x−μ)22σ2),CDF(x)=12(1+erf(x−μσ2–√)),Quantile(p)=μ+σ2–√erf−1(2p−1);Pr(|X−μ|<σX)≈68.27%,Pr(|X−μ|<2σX)≈95.45%,Pr(|X−μ|<3σX)≈99.73%;
Exponential distribution: x≥0,PDF(x)=λe−λx,CDF(x)=1−e−λx,Quantile(p)=−ln(1−p)λ;μ=1λ,σ2=1λ2;
Binomial distribution: Pr(X=k)=(nk)pk(1−p)n−k;μ=np,σ2=np(1−p);
Geometric distribution with k≥1: Pr(X=k)=(1−p)k−1p,Pr(X≤k)=1−(1−p)k;μ=1p,σ2=1−pp2;
Geometric distribution with k≥0: Pr(X=k)=(1−p)kp,Pr(X≤k)=1−(1−p)k+1;μ=1−pp,σ2=1−pp2;
Poisson distribution: PDF(k)=λke−λk!;μ=λ,σ2=λ;
*Transformations of probability density functions
Probability density function pX(x) transformed using y=y(x): pY(y)=pX(x(y))∣∣∣dxdy∣∣∣; paX+b(y)=1|a|pX(y−ba);
Convolution of probability distributions: let Z=X+Y,Pr(Z=z)=∑kPr(X=k)Pr(Y=z−k),pZ(z)=∫∞−∞pX(z−t)pY(t)dt=∫∞−∞pX(t)pY(z−t)dt;
∑i=1nNormal(μi,σ2i)∼Normal(∑i=1nμi,∑i=1nσ2i);∑i=1nBinomial(ni,p)∼Binomial(∑i=1nni,p);∑i=1nPoisson(λi)∼Poisson(∑i=1nλi);
Miscellaneous Algebra
a3±b3=(a±b)(a2∓ab+b2),a4−b4=(a−b)(a3+a2b+ab2+b3),an−bn=(a−b)(an−1+an−2b+an−3b2+⋯+a2bn−3+abn−2+bn−1);
1ab=1b−a(1a−1b);11x+1y=xyx+y,11x+1y+1z=xyzxy+xz+yz;11−x−11+x=2x1−x2;
Solutions of ax2+bx+c=0: x=±b2−4ac−−−−−−−√−b2a; Solutions of ax2+2bx+c=0: x=±b2−ac−−−−−−√−ba;
Vertex of ax2+bx+c: (−b2a,4ac−b24a); Inflection point of ax3+bx2+cx+d: (−b3a,27a2d−9abc+2b327a2);
Let x1, x2 be the roots of ax2+bx+c: x1+x2=−ba,x1⋅x2=ca,x21+x22=b2−2aca2;
Parabola through (x1,y1),(x2,y2),(x3,y3): f(x)=y1⋅(x−x2)(x−x3)(x1−x2)(x1−x3)+y2⋅(x−x1)(x−x3)(x2−x1)(x2−x3)+y3⋅(x−x1)(x−x2)(x3−x1)(x3−x2);
f(x)=ax2+bx+c,a=x1(y3−y2)+x2(y1−y3)+x3(y2−y1)(x1−x2)(x1−x3)(x2−x3)=y1(x2−x3)+y2(x3−x1)+y3(x1−x2)(x1−x2)(x1−x3)(x2−x3),b=−x21(y3−y2)+x22(y1−y3)+x23(y2−y1)(x1−x2)(x1−x3)(x2−x3)=−y1(x22−x23)+y2(x23−x21)+y3(x21−x22)(x1−x2)(x1−x3)(x2−x3),c=x21(x2y3−x3y2)+x22(x3y1−x1y3)+x23(x1y2−x2y1)(x1−x2)(x1−x3)(x2−x3)=y1(x2x3(x2−x3))+y2(x3x1(x3−x1))+y3(x1x2(x1−x2))(x1−x2)(x1−x3)(x2−x3);
Sequence and Series
Arithmetic Sequence: an=a1+d(n−1), n=an−a1d+1, d=an−a1n−1; Sn=n(a1+an)2=na1+n(n−1)2d, Pn=dnΓ(a1/d+n)Γ(a1/d)(a1/d∉N−);
Geometric Sequence: an=a1⋅qn−1, n=logqana1+1, q=(ana1)1n−1; Sn=a1(qn−1)q−1=anq−a1q−1; Pn=an1⋅qn(n−1)2;
∑k=0∞rk=⎧⎩⎨⎪⎪⎪⎪⎪⎪+∞11−rdiverger≥1|r|<1r≤−1; ∑k=1nrk=r(1−rn)(1−r),∑k=0nrk=1−rn+11−r;
∑k=1nk=12n(n+1),∑k=1nk2=16n(n+1)(2n+1),∑k=1nk3=14n2(n+1)2,∑k=1nk4=130n(n+1)(2n+1)(3n2+3n−1);
(a+b)n=∑k=0n(nk)akbn−k=∑k=0n(nk)an−kbk,(1+x)n=∑k=0n(nk)xk,(1−x)n=∑k=0n(nk)(−1)kxk,(x−1)n=∑k=0n(nk)(−1)n−kxk;
Taylor and Maclaurin Series
11−x=∑k=0∞xk=1+x+x2+x3+⋯,11+x=∑k=0∞(−1)kxk=1−x+x2−x3+⋯,ln(1+x)=∑k=0∞(−1)kxk+1k+1=x−x22+x33−x44+⋯;(|x|≤1)
11−x2=∑k=0∞x2k=1+x2+x4+x6+⋯,11+x2=∑k=0∞(−1)kx2k=1−x2+x4−x6+⋯,
arctan(x)=∑k=0∞(−1)kx2k+12k+1=x−x33+x55−x77+⋯,arctanh(x)=∑k=0∞x2k+12k+1=x+x33+x55+x77+⋯;(|x|≤1)
ex=∑k=0∞xkk!=1+x+x22!+x33!+⋯,sin(x)=∑k=0∞(−1)kx2k+1(2k+1)!=x−x33!+x55!−x77!+⋯,cos(x)=∑k=0∞(−1)kx2k(2k)!=1−x22!+x44!−x66!+⋯,
sinh(x)=∑k=0∞x2k+1(2k+1)!=x+x33!+x55!+x77!+⋯,cosh(x)=∑k=0∞x2k(2k)!=1+x22!+x44!+x66!+⋯;
Fourier Series and Integral
Function f with period T: f(t)=a02+∑k=1∞(ancos(kωt)+bnsin(kωt)), where ak=2T∫t0+Tt0f(t)cos(kωt)dt,bk=2T∫t0+Tt0f(t)sin(kωt)dt,ω=2πT;
or f(t)=∑k=−∞∞cnekiωt,ck=1T∫t0+Tt0f(t)e−kiωtdt,ω=2πT; ck=c¯¯−k for real f;
When T=2π: f(x)=a02+∑k=1∞(akcos(kx)+bksin(kx)),ak=1π∫π−πf(x)cos(kx)dx,bk=1π∫π−πf(x)sin(kx)dx; 1π∫π−πf(x)2dx=a202+∑k=1∞(a2k+b2k)
Fourier integral: f(x)=12π∫∞−∞dλ∫∞−∞f(t)e−iλ(t−x)dt=12π∫∞−∞C(λ)eiλxdλ,C(λ)=∫∞−∞f(t)e−iλtdt;
or f(x)=1π∫∞0(A(λ)cosλx+B(λ)sinλx)dλ,A(λ)=∫∞−∞f(t)cosλt dt,B(λ)=∫∞−∞f(t)sinλt dt;
* Fourier Transform
Fourier transform and its inverse: f^(ξ)=∫∞−∞f(x)e−2πiξxdx,f(x)=∫∞−∞f^(ξ)e2πiξxdξ;
F{af(x)+bg(x)}=af^(ξ)+bg^(ξ),F{f(x−x0)}=e−2πiξx0f^(ξ),F{e2πiξ0xf(x)}=f^(ξ−ξ0),F{f(ax)}=1|a|f^(ξa)(a∈R),F{f(−x)}=f^(−ξ),F{f(x)¯¯¯¯¯¯¯¯¯¯}=f^(−ξ)¯¯¯¯¯¯¯¯¯¯¯¯¯¯;f^(−ξ)=f^(ξ)¯¯¯¯¯¯¯¯¯¯(f∈R);
F{ddxf(x)}=2πiξ⋅f^(ξ),F{dndxn}=(2πiξ)nf^(ξ);*F{−2πixf(x)}=ddξf^(ξ);
*F{(f∗g)(x)}=f^(ξ)⋅g^(ξ),F{f(x)⋅g(x)}=(f^∗g^)(ξ); where (f∗g)(x)=∫∞−∞f(t)g(x−t)dt;
*∫∞−∞|f(x)|2dx=∫∞−∞∣∣f^(ξ)∣∣2dξ,∫∞−∞f(x)g(x)¯¯¯¯¯¯¯¯¯dx=∫∞−∞f^(ξ)g^(ξ)¯¯¯¯¯¯¯¯¯dξ;(∫∞−∞|f(x)|2dx<∞, ∫∞−∞|g(x)|2dx<∞;)
Discrete Fourier Transform
Discrete Fourier transform and its inverse: Xk=∑n=0N−1xnexp(−2πiknN),xn=1N∑k=0N−1Xkexp(2πinkN);
F({axn+byn})k=aXk+bYk,F({x−n})k=X−k,F({xn¯¯¯¯¯})k=X−k¯¯¯¯¯¯¯¯¯,F({xn−t})k=e−2πitkNXk,F({e2πitnNxn})k=Xk−t;
F−1({XkYk})n=∑t=0N−1xt⋅yn−t,F({xnyn})k=1N∑t=0N−1xt⋅yk−t;
∑n=0N−1xnyn¯¯¯¯¯=1N∑k=0N−1XkYk¯¯¯¯¯,∑n=0N−1|xn|2=1N∑k=0N−1|Xk|2;
