SMall Is Beautiful

http://sourceforge.net/projects/smib/

philippe.billet@noos.fr

1 Introduction

• number theory (integer, primality, arithmetic functions, Dirichlet convolution),
• algebra (polynomial, matrix, ...),
• analysis (trigonometry, special functions, differential operator, integration, Fourier transform, Taylor sum),
• differential geometry (curves, surfaces, tensorial calculus),
• numerical analysis (interpolation, numerical integration, ordinary differential equation, sample manipulation, numerical Fourier transform, differintegration, complex analysis),
• probality & statistic (expected value, variance, standard deviation, skewness, kurtosis, least square fitting, quantile, stochastic differential equation).

2 Compilation & use

↓Compilation

• make
• gcc
• g++

Use

• Interactive mode↓↓: open a terminal, and type ./smib (if you are in source folder, or smib if you copy executable), then you have the prompt messages, it’s up to you.
• ↓↓Script mode: here you must have a valid file (there are some examples in folder /smib/documentation and /smib/documentation/application - in folder /smib/documentation/tutorial gives basics examples). Then type ./smib ./documentation/tutorial, smib treats the file. N.B.: if file ends with quit() then, after treating, smib goes back to shell, else it enters in interactive mode.
• In source directory, the file init.cpp contains many user defined functions, almost all are discribed in this document in section 5 Applications.

Boolean indicators

3 Mathematical objects

3.1 Numbers

↓Integers

How to define

• ↓Mersenne number↓: ↓mersenneN(n)
• ↓Fermat number↓: ↓fermatN(n)
• ↓Fibonacci number↓: ↓fibonacciN(n)
• ↓Catalan number↓: ↓catalanN(n)
• ↓↓↓Stirling numbers↓ of first and second kind: ↓stirlingN1k(n,m), ↓stirlingN2k(n,m)
• ↓↓Bell number: ↓bellN(n)
• ↓↓Genocchi number: ↓genocchiN(n)
• ↓↓↓↓Euler numbers of first and second kind: ↓eulerN1k(n), ↓eulerN2k(n,m)

↓Functions acting on integers

• odd↓ or even↓ - odd: ↓isodd(n); even: ↓iseven(m)
• ↓↓least common multiple & great common divisor↓↓ - gcd: ↓gcd(5,3), lcm: ↓lcm(5,3)
• counting: factorial↓: 2!↓, binomial coefficient↓: ↓binomial(5,3), ↓permutation: ↓permutation(5,3), ↓derangement: ↓derangement(5)
• integer factorization & primality test: factorization↓: ↓factor(n), ↓primedecomp(n) returns 2 vectors the first contains prime factors, the second exponents of corresponding factor; primality test↓: ↓isprime(n)
• number theory functions & operators:
  • operators acting on arithmetic function:
    sum over divisors↓↓ (∑_d|Nf(d)): ↓sumoverdivisor(x,N,f),
    sum over primes↓↓ (∑_p|Nf(p)): ↓sumoverprime(x,N,f),
    product over divisors↓↓ (∏_d|Nf(d)): ↓prod uctoverdivisor(x,N,f),
    product over primes↓↓ (∏_p|Nf(p)): ↓productoverprime(x,N,f),
    Dirichlet convolution product↓↓ (f⋆g(n) = ∑_d|nf(d)g(ⁿ⁄_d)): ↓dirichletproduct(f,g,x,n)
  • functions:
    number of distinct prime factors↓:↓omega(n)
    total number of prime factors↓: ↓Omega(n)
    number of divisors↓ ↓: ↓tau(n)
    prime counting↓: ↓pi(n)
    sum of divisors at ordre r↓: ↓sigma(n,r)
    kernel of integer↓ ↓: ↓integerkernel(n)
    ↓Liouville function↓: ↓liouvillefunction(n)
    ↓Euler totient function↓: ↓eulerphi(n)
    ↓↓Möbius function↓: mobius(n)
    ↓van Mangoldt function↓: ↓mangoldt(n)
    Legendre symbol↓↓: ↓legendresymbol(n,m)
    Jacobi symbol↓↓: ↓jacobisymbol(n,m)

↓Rational numbers

How to define

• ↓Bernoulli number↓: ↓bernoulliN(n)

Functions acting on rational numbers

• ↓numerator(q) returns numerator↓ of a rational
• ↓denominator(q) returns denominator↓ of a rational
• ↓rationalize(q) ↓reduces of rational expression

↓Real numbers

How to define

• ↓Archimedes constant π↓: pi↓,
• ↓unity exponential e: exp(1),
• ↓↓Euler constant γ: ↓euler.

Functions acting on real numbers

Complex numbers↓

How to define

Functions acting on complex numbers

• ↓real(z) returns real part↓ of z
• ↓imag(z) returns ↓imaginary part of z
• ↓abs(z) returns absolute value↓ of z
• ↓arg(z) returns ↓argument of z
• ↓conj(z) returns conjugate↓ of z

Functions acting on any type of numbers

Variables

How to define

Functions acting on variables

3.2 Structures of higher dimensions

↓Vectors

How to define

Functions acting on vectors

↓Matrices

How to define

Functions acting on matrices

↓Polynomials

How to define

• ↓Cyclotomic polynomial↓: ↓cyclotomic(x,n)
• ↓Orthogonal polynomials↓:
  • ↓Hermite polynomials↓: ↓hermite(x,n)
  • ↓Laguerre polynomials↓: ↓laguerre(x,n,k)
  • ↓↓Legendre polynomials: ↓legendre(x,n,m)
  • U & T ↓Tchebychev polynomials↓: ↓tchebychevU(x,n) & ↓tchebychevT(x,n).

Functions acting on polynomials

↓Samples

How to define

Functions acting on samples

↓Tensors

• ↓↓covariant derivative: T^α_;γ = T^α_, γ − T^μΓ^α_μγ
• ↓commutator: [u, v] = (u^βv^α_, β − v^βu^α_, β)e_α
• ↓↓directed covariant derivative: ∇_uv = v^α_;βu_α(using that we have too:[u, v] = ∇_vu − ∇_uv)
• ↓↓covariant derivatives of tensor:
  • T^αβ_;γ = T^αβ_, γ + T^μβΓ^α_μγ + T^αμΓ^β_μγ
  • T^α_β;γ = T^α_β, γ + T^μ_βΓ^α_μγ − T^α_μΓ^μ_βγ
  • T_αβ;γ = T_αβ, γ − T_μβΓ^μ_αγ − T_αμΓ^μ_βγ

3.3 Operators & functions

Functions

Operators

4 Elements of language

↓prog

↓Condition

• ↓equality: ==
• ↓inequality: < or >
• ↓isinteger
• ↓logic disjonction: or
• ↓logic conjonction: and
• ↓logic negation: not

↓Loop

• ↓for(variable,initial value, final value,action to do), for is very practice for every indiced objects (vectors, matrices,...)
• recursive loop↓: an example is worth a thousand words gcd of two polynomials:
  gcdpoly(a,b,x)=prog(temp1,
  do(test(degree(a)<degree(b),
  gcdpoly(b,a,x),
  do(test(b==0,a,gcdpoly(b,divpoly(a,b,x)[2],x))))))

↓nil↓

↓Plotting

5 Applications

• arithmetic & number theory:
  • primality tests
  • arithmetic functions properties
  • Dirichlet’s product properties
  • π computation
• function manipulation applied to quantum mechanic:
  • operator (creation, annihilation, commutator...)
  • operator acting on function
  • relation between quantum mechanic and Hermite function
• differential geometry:
  • theory of curves: planar & 3D
  • gaussian theory of surfaces
  • riemannian geometry
• unidimensional calculus:
  • classical integrals
  • orthogonal polynomial properties
  • complex path integrals
  • continous Fourier transform:
    • elementary functions
    • Fourier transform & convolution product
    • Fourier transform & Green functions
• vectorial calculus:
  • vectorial integrals
  • curvilinear integrals
  • area integrals
  • Green theorem
  • Stokes theorem
  • Gauss-Ostrogradski theorem
• numerical analysis:
  • univariate calculus applied to samples
  • special function (Bessel, Hankel, Airy...)
  • interpolation (Lagrange-Newton, least squares mean)
  • polynomial interpolation
  • numerical integration
  • discrete Fourier transform using samples:
    • elementary functions
    • differential equations
    • pseudo-differential operator
    • wavelets
  • complex analysis
  • ordinary differential equation
  • fractional calculus
• probability & statistic:
  • distributions
  • moments
  • quantile & median
  • random numbers generation
  • law of large numbers & central limit theorem
  • stochastic differential equation

5.1 Arithmetic & Number theory

5.1.1 Computing famous numbers

Using direct computation

Using recursive computation

Using generating function

5.1.2 ↓primality tests

↓Deterministic test: using ↓Wilson formula↓

↓Probabilistic test: ↓Solovay-Strassen test

5.1.3 properties of arithmetic functions↓

↓Sum over divisors

• ∑_d|N|μ(d)|φ(d) = γ(N)
• ∑_d|N¹⁄_d² = ^σ(N, 2)⁄_N²
• ∑_d|N|μ(d)| = 2^ω(N)
• ∑_d|Nμ(d)τ(d) = ( − 1)^ω(N)
• ∑_d|Nμ(d)λ(d) = 2^ω(N)
• ∑_d|Nσ(d, 1) = N.∑_d|N^τ(d)⁄_d

Product over divisors↓ (or prime divisors↓)

• ∏_d|Nd = N.^τ(N)⁄₂
• ( − 1)^ω(N)∏_p|Np = ∑_d|Nμ(d)σ(d, 1)
• ( − 1)^ω(N)∏_p|N(p − 2) = ∑_d|Nμ(d)φ(d).

↓Dirichlet product

• σ = id⋆1
• τ = 1⋆1
• φ = μ⋆id
• σ = φ⋆τ
• id = σ⋆μ
• id = φ⋆1
• id.τ = φ⋆σ
• (id.σ)(n) = ∑_d|nτ(d)
• (id².σ)(n) = ∑_d|nd.σ(d)
• σ⋆σ = (id.τ)⋆τ
• ∑_1≤n≤N(τ⋆μ⋆Λ)(n) = log(N!)(here X = (1, 2, 3, 4, 20, 25, 154))
  

5.1.4 ↓↓Arithmetic equations

↓Bombieri conjecture

4τ(n + 2) = φ(n),  0≤n≤1000

σ(n, 1) = 2n + 1,  0≤n≤1000

5.1.5 Arithmetic & statistic

• statkurtosis(listprime(n)) and ku(n), and smib gives:

• statskewness(listprime(n)) and sk(n), and smib gives:

5.1.6 ↓↓Gould number

5.1.7 ↓π computation (cooking π in ten lessons)

1. ↓↓Wallis formula: π = 2∏^j = ∞_j = 1^(2j2)⁄_{(2j − 1)(2j + 1)}
2. From Wallis formula: π = ∑^j = ∞_j = 0^{2j + 1j!2}⁄_{(2j + 1)!}
3. Unknown man formula: π = 4(1 − ∑^j = ∞_j = 1²⁄_{16j² − 1})
4. ↓Wang formula↓: π = 4(183arctan(¹⁄₂₃₉) + 32arctan(¹⁄₁₀₂₃) − 68arctan(¹⁄₅₈₃₂) + 
   12arctan(¹⁄₁₁₀₄₄₃) − 12arctan(¹⁄_4841182) − 100arctan(¹⁄_6826318))
5. Using ζ ↓↓Riemann function: π = √(6∑^j = ∞_j = 1¹⁄_j²)
6. Using sequence of prime numbers↓↓: π = √(6∏_j ∈ P¹⁄_{1 − ¹⁄j²}), where P is the set of prime numbers.
7. ↓↓Ramanujan formula: ¹⁄_π = ^2√(2)⁄₉₈₀₁∑^j = ∞_j = 1^{(4j)!(1103 + 26390j)}⁄_{(j!⁴396^4j)}
8. ↓Chudnowsky formula↓: ¹⁄_π = 12∑^j = ∞_j = 0( − 1)^j(6j)!^{13591409 + 545140134j}⁄_{((3j)!j!³640320^{(3(j + 1 ⁄ 2))})}
9. ↓↓Borwein-Borwein-Plouffe formula: π = ∑^j = ∞_j = 016^− j[⁴⁄_8j + 1 − ²⁄_8j + 4 − ¹⁄_8j + 5 − ¹⁄_8j + 6]
10. ↓↓Bellard formula: π = ∑^j = ∞_j = 0^{( − 1)j}⁄_2^10j[^− 25⁄_4j + 1 − ¹⁄_4j + 3 + ²⁸⁄_10j + 1 − ²⁶⁄_10j + 3
     − ²²⁄_10j + 5 − ²²⁄_10j + 7 + ¹⁄_10j + 9]

5.2 Function manipulation applied to quantum mechanic↓

5.2.1 ↓Operator definition

• ↓creation: a⁺(f)(x) = ^{xf(x) − df(x)⁄_dx}⁄_√(2),
• ↓annihilation: a(f)(x) = ^{xf(x) + df(x)⁄_dx}⁄_√(2),
• ↓position: X(f)(x) = ^{√(2)(a(f(x)) + a + (f, x))}⁄₂,
• ↓impulsion: P(f)(x) = ^{i√(2)( − a(f)(x) + a + (f)(x))}⁄₂,
• ↓quanta number: N(f)(x) = a⁺(a(f))(x),
• ↓hamiltonian: H(f)(x) = N(f)(x) + ^f(x)⁄₂,
• ↓commutator: [a₁, a₂](f, x) = (a₁○a₂ − a₂○a₁)(f)(x).
  

5.2.2 ↓Operator acting on function

5.2.3 Comparison with analytic solution

5.3 Differential geometry↓

5.3.1 Curves

↓↓Planar curves

↓3D curves

5.3.2 ↓Theory of surfaces

• ↓↓first form, and its determinant↓,
• ↓↓normal vector,
• ↓↓second form,
• ↓↓Weingarten operator,
• ↓↓Mean curvature, Gaussian curvature↓↓, and principal curvature↓↓,
• ↓Shape index↓ and curvedness↓.

5.3.3 ↓Riemannian geometry

5.4 Unidimensional integral↓ calculus↓

5.4.1 Classical integrals

• ∫_{[ − ∞, ∞]}1_{[ − 3, 9]}dx
• ∫_{[ − ∞, ∞]}e^− x2dx
• ∫_{[ − ∞, ∞]}¹⁄_{(1 + x²)}dx
• ∫_[0, ∞]^sin(x)⁄_xdx
• ∫_{[ − ∞, ∞]}^sin(x)⁄_xdx
• ∫_[0, ∞]e^− xdx.

5.4.2 ↓↓Orthogonal polynomial properties

5.4.3 ↓↓Complex path integrals

5.4.4 ↓Continous Fourier transform↓

• On functions:
  a constant: print("fourier(a,t)=",fourier(a,t))
  sin(t): print("fourier(sin(t),t)=",expand(fourier(sin(t),t)))
  e^− a|t|: print("fourier(exp(-a*abs(t)),t)=",fourier(exp(-a*abs(t)),t))
  log(a|t|): print("fourier(log(a*abs(t)),t)=",fourier(log(a*abs(t)),t))
  e^− t2: print("fourier(exp(-t^2),t)=",fourier(exp(-t^2),t))
  e^it2: print("fourier(exp(i*t^2),t)=",fourier(exp(i*t^2),t))
  arctan(t): print("fourier(arctan(t),t)=",fourier(arctan(t),t))
  erf(t): print("fourier(erf(t),t)=",fourier(erf(t),t))
  
• On operators:
  ∂²_xf: print("fourier(d(f(t),t,2),t)=",fourier(d(f(t),t,2),t))
  f⋆g: print("fourier(convolution(f(t),g(t)),t)=",
              expand(fourier(convolution(f(t),g(t)),t)))
  ∫f: print("fourier(integral(f(t),t),t)=",fourier(integral(f(t),t),t))
• Applications:
  ℱ^− 1(¹⁄_{ℱ(^dδ(t)⁄dt − δ(t))}):
  print("invfourier(1/fourier(d(dirac(t),t)-dirac(t),t),t)=",
  invfourier(1/fourier(d(dirac(t),t)-dirac(t),t),t))
  print("d(",last,",t)-",last,"=",d(last,t)-last)
  ℱ^− 1(¹⁄_{ℱ(t^dδ(t)⁄dt − δ(t))}):
  print("invfourier(1/fourier(t*d(dirac(t),t)-dirac(t),t),t)=",
  invfourier(1/fourier(t*d(dirac(t),t)-dirac(t),t),t))
  print("t*d(",last,",t)-",last,"=", t*d(last,t)-last)
  t∂_tδ (it is just an application of the following propertie (f.g = ℱ^− 1(ℱ(f)⋆F(g)):
  print("fourierprod(d(dirac(t),t),t)=", expand(fourierprod(d(dirac(t),t),t)))

• f⋆sgn = 2∫f:
  print("convolution(f(x),sgn(x))=", expand(convolution(f(x),sgn(x))))
• ¹⁄_x⋆¹⁄_x =  − π²δ(x):
  print("convolution(1/x,1/x)=", expand(convolution(1/x,1/x)))
• ¹⁄_x²⋆¹⁄_x² =  − π^2dδ(x)⁄_dx:
  print("convolution(1/x^2,1/x^2)=", expand(convolution(1/x^2,1/x^2)))
• e^− t2⋆e^− t2 = √(^π⁄₂)e^{− t2⁄₂}:
  print("convolution(exp(-t^2),exp(-t^2))=",
  expand(convolution(exp(-t^2),exp(-t^2))))

• (H): ∂_tf − ∂²_xf:
  d(f(x,t),t)-d(f(x,t),x,2)
  fourier(last,x)
  expand(last)
  dsolve(last,fourier(f(x,t),x),t)
  invfourier(last,x)
  d(last,t)-d(last,x,2) (I)
  
  We have the following symbolic solution:
  
  If we sample (I), we can see that it is all zero: symbolic solution is correct.
• (S): ∂_tf − i∂²_xf:
  d(f(x,t),t)-i*d(f(x,t),x,2)
  fourier(last,x)
  expand(last)
  dsolve(last,fourier(f(x,t),x),t)
  invfourier(last,x)
  d(last,t)-i*d(last,x,2) (II)
  
  We have the following symbolic solution:
  
  For (II) as for (I), the sample is all zero: symbolic solution is correct.

5.5 ↓Vectorial calculus

5.5.1 ↓↓Vectorial integrals

5.5.2 ↓↓Curvilinear integrals

• Let C = {x(t) = 2t, t ∈ [0, 1]} a planar curve, andy(t) = 2t a function, we want to compute ∫_Cy²ds. So in smib:
  
  t=quote(t)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  x=2t
  y=2t
  print("defint(y^2*d(x,t),t,0,1)=",defint(y^2*d(x,t),t,0,1))
  
  And smib gives:
  
  defint(y^2*d(x,t),t,0,1)= 8/3
  
• Let C = {(x(t) = 2t + 1, y(t) = t², z(t) = 1 + t³), t ∈ [0, 1]} a curve in ℝ³, and A⃗ = (z, x, y) a vector, we want to compute ∫_CA⃗.ds. So in smib:
  
  t=quote(t)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  x=2t+1
  y=t^2
  z=1+t^3
  f=z*d(x)+x*d(y)+y*d(z)
  print("defint(z*d(x)+x*d(y)+y*d(z),t,0,1)=",defint(f,t,0,1))
  
  And smib gives:
  
  defint(z*d(x)+x*d(y)+y*d(z),t,0,1)= 163/30
  
• Let C = {(x(t) = t² + 1, y(t) = 2t², z(t) = t³), t ∈ [1, 2]} a curve in ℝ³, and a vector F⃗ = (3xy,  − 5z, 10x), we want to compute ∫_CF⃗.ds. So in smib:
  
  t=quote(t)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  F=quote(F)
  G=quote(G)
  C=quote(C)
  x=t^2+1
  y=2*t^2
  z=t^3
  F=(3*x*y,-5*z,10*x)
  C=(x,y,z)
  print("defint(dotP(F,d(C,t)),t,1,2)=",defint(dotP(F,d(C,t)),t,1,2))
  
  And smib gives:
  
  defint(dotP(F,d(C,t)),t,1,2)= 303
  
• Let C = {(x(t) = t², y(t) = 2t, z(t) = t³), t ∈ [1, 2]} a curve in ℝ³,φ = 2xyz² a function and F⃗ = (xy,  − z, x²) a vector, we want to compute ∫_Cφds and ∫_CF⃗∧ds. So in smib:
  
  t=quote(t)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  F=quote(F)
  phi=quote(phi)
  G=quote(G)
  C=quote(C)
  x=t^2
  y=2*t
  z=t^3
  phi=2*x*y*z^2
  F=(x*y,-z,x^2)
  C=(x,y,z)
  print("defint(phi*d(C,t),t,0,1)=",defint(phi*d(C,t),t,0,1))
  print("defint(crossP(F,d(C,t)),t,0,1)=",defint(crossP(F,d(C,t)),t,0,1))
  
  And smib gives:
  
  defint(phi*d(C,t),t,0,1)= (8/11,4/5,1)
  defint(crossP(F,d(C,t)),t,0,1)= (-9/10,-2/3,7/5)

5.5.3 Area integrals↓↓

• Let S = {(x, y, x² + 2y), (x, y) ∈ [0, 1]²} a map in ℝ³, we want to compute ∬_SdS. So in smib:
  
  a=quote(a)
  t=quote(t)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  z=x^2+2y
  S=(x,y,z)
  a=abs(crossP(d(S,x),d(S,y)))
  print("S=",S)
  print("defint(defint(abs(crossP(d(S,x),d(S,y))),x,0,1),y,0,1)=",
  defint(defint(a,x,0,1),y,0,1))
  
  And smib gives:
  
  S= (x,y,2*y+x^2)
  
  defint(defint(abs(crossP(d(S,x),d(S,y))),x,0,1),y,0,1)=
  3/2-20*log(2)-10*log(5)+20*log(10)
  
• Let S = {(ucos(v), usin(v), v), (u, v) ∈ [0, 1] × [0, 3π]} a map in ℝ³, we want to compute ∬_SdS. So in smib:
  
  a=quote(a)
  t=quote(t)
  u=quote(u)
  v=quote(v)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  x=u*cos(v)
  y=u*sin(v)
  z=v
  S=(x,y,z)
  a=abs(crossP(d(S,u),d(S,v)))
  a=simplify(a)
  print("S=",S)
  print("defint(defint(abs(crossP(d(S,u),d(S,v))),u,0,1),v,0,3pi)=",
  defint(defint(a,u,0,1),v,0,3pi))
  
  And smib gives:
  
  S= (u*cos(v),u*sin(v),v)
  
  defint(defint(abs(crossP(d(S,u),d(S,v))),u,0,1),v,0,3pi)=
  -3/2*pi*log(2)+3/2*pi*log(2+2*2^(1/2))+3*pi/(2^(1/2))

5.5.4 Some famous theorems

1. Let S a map in ℝ³defined by S = {(x, y, 4 − x² − y²), (x, y) ∈ [ − 1, 1]²}, and ⟶F = (y, z, x)
2. Let S a map in ℝ³ defined by S = {(x, y, √(1 − x² − y²)), (x, y) ∈ [ − 1, 1]²}, and ⟶F = (2x − y,  − x²y,  − y²z).

5.6 ↓Numerical analysis

5.6.1 Sample↓s applied to numerical analysis

5.6.2 Special functions

5.6.3 Numerical interpolation

5.6.4 Numerical integration

↓Integrals on compact

• ∫¹₀√(x)dx, ∫¹₀x^3 ⁄ 2dx
• ∫¹₀¹⁄_1 + xdx
• ∫¹₀¹⁄_1 + x⁴dx
• ∫¹₀¹⁄_1 + e^xdx
• ∫¹₀^x⁄_{− 1 + e^x}dx
• ∫¹₀²⁄_{2 + sin(10πx)}dx
• ∫¹₀¹⁄_{x^1⁄2 + x^1⁄3}dx.

↓↓Oscillatory integrals

• I₀(n) = ∫^2π₀xcos(x)sin(nx)dx
• I₁(n) = ∫^2π₀xcos(50x)sin(nx)dx
• I₂(n) = ∫^2π₀^xsin(nx)⁄_{√(1 − (^x⁄2π)²)}dx
• I₃(n) = ∫^2π₀ln(x)sin(nx)dx.
  

Integrals on semi-finite interval↓

• ∫^∞₀^e − x⁄_1 + x⁴dx
• ∫^∞₀e^− x2dx
• ∫^∞₀^e − x⁄_2x + 100dx
• ∫^∞₂¹⁄_xln(x)²dx
• ∫^∞₂¹⁄_xln(x)^3⁄2dx
• ∫^∞₂¹⁄_x^1.01dx
• ∫^∞₂^sin(x)⁄_xdx
• ∫^∞₂cos(π^x2⁄₂)dx
• ∫^∞₂e^− x2dx
• ∫^∞₂^{sin(x − 1)}⁄_{√(x(x − 2))}dx
• ∫^∞₂^x⁄_{e^x − 1}dx.
  

↓Oscillatory integrals on semi-finite interval

• I(w) = ∫^∞₀e^− xsin(wx)dx
• J(w) = ∫^∞₀sin(wx)^x⁄_1 + x²dx
• K(w) = ∫^∞₀sin(wx)^{sin(x⁄₂)2}⁄_xdx
• ∫^∞₀10e^{− xsin(16πx)}⁄_1 + x²dx.
  

↓↓Miscellanous integrals

• ∫¹₀e^{(x2(1 − x2))}dx
• I(t, n) = ∫^π₀^{cos(tsin(x) − nx)}⁄_πdx
  

↓Wavelets

5.6.5 ↓↓Discrete Fourier transform

• i^df(t)⁄_dt + f(t) = δ(t) (I)
• ^d2f(t)⁄_dt² − f(t) = δ(t) (II)

5.6.6 Complex analysis↓

5.6.7 ↓Ordinary differential equation