Calculus

  1. Taylor:Taylor
    • series(cos(x),x,0,5) >>>1-x22+x424+O(x5)

  2. Gamma:Γ(x)=0tx-1etdt,x>0
    • Γ(1) >>>1

    • gamma(1) >>>1

  3. Zeta:ζ(s)=lim(1+12s+13s+14s...+1ns)
    • ζ(2) >>>1.64493406684823

    • zeta(2) >>>1.64493406684823

  4. Limit:limxy
    • limx0sin(x)x >>>1

    • limit(sin(x)/x,x,0) >>>1

  5. Right limit:limxy+
    • limx0+1x >>>oo

    • limit(1/x,x,0,'+') >>>oo

  6. Left limit:limxy-
    • limx0-1x >>>-oo

    • limit(1/x,x,0,'-') >>>-oo

  7. Indefinite integral:
    • xdx >>>0.5*x2

    • integrate(x,x) >>>0.5*x2

  8. Definite integral:ab
    • 0e-xxdx >>>1

    • integrate(pow(e,-x),x,0,inf) >>>1

  9. Multiple integral

    Repeat the corresponding integral operation.

  10. N-th order derivative: dnydxn
    • dln(x)dx >>>x-1

    • diff(ln(x),x,1) >>>x-1

  11. N-th order derivative:y(n)
    • y=ln(x) y'(x) >>>x-1

    • diff(ln(x),x,2) >>>-1x2

  12. N-th order partial derivative:nyxn
    • y*ln(x)∂x >>>yx

    • diff(y*ln(x),x,2) >>>-yx2

  13. Heaviside step function:u(t)

    u(t)= 1x>0 12x=0 0otherwise

  14. Fourier transform:FT

    F(k)= FT(f,x,k)= -f(x)e-2πixkdx

    • F= FT(e-x2,x,k) >>>F=π* e-π2*k2

  15. Inverse Fourier transform:FT-1

    f(x)= FT-1(F,k,x)= -F(k)e2πixkdk

    • F= FT(e-x2,x,k) FT-1(F,k,x) >>>e-x2

  16. Laplace transform:LT

    F(s)= LT(f,t,s)= 0e-stf(t)dt

    • F= LT(1,t,s) >>>F=1s

  17. Inverse Laplace transform:LT-1

    f(t)= LT-1(F,s,t)= c-ic+ie-stF(s)ds

    • F= LT(1,t,s) LT-1(F,s,t) >>>u(t)

  18. Mellin transform:MT

    F(s)= MT(f,t,s)= 0xs-1f(x)dx

    • F= MT(e-t,t,s) >>>F=Γ(s)

  19. Inverse Mellin transform:MT-1

    f(t)= MT-1(F,s,t)= c-ic+ix-sF(s)ds

    • F= MT(e-t,t,s) MT-1(F,s,t) >>>e-t