Exit[]

Play[(3-t)Sin[600 2 Pi t]+t Sin[600 2 2Pi t],{t,0,3}]

pB=Sin[Pi^(7/2)+ I]

expF=Sin[Pi^SetPrecision[3.5,50]+ I]

(expF-pB)/pB

expG=Sin[Pi^N[3.5,50]+ I]

(expG-pB)/pB

expG=Sin[N[Pi^3.5,50]+ I]

(expG-pB)/pB

sqsqrt[x_,n_]:=
Nest[#^2&,Nest[Sqrt,x,n],n]

sqrtsq[x_,n_]:=
Nest[Sqrt,Nest[#^2&,x,n],n]

{sqrtsq[2.5,3],sqsqrt[2.5,3]}

n=28;
Map[{#,(#-sqsqrt[#,n])/#,(#-sqrtsq[#,n])/#}&,
{.1,.5,1/2,.9,1,1.,2,Pi,t}]//TableForm

n=25;
Map[{#,sqsqrt[#,n],sqrtsq[#,n]}&,
{.1,.5,1/2,.9,1.1,2,Pi,t}]//TableForm

Nest[Sqrt,.1,30]

Nest[#^2&,1.1,13]

N*

Difference between the Mathematica functions "N*" and "*": numeric vs algebraic solvers!
Solve[...]//N is NOT the same as NSolve[...]. Same for Sum vs. NSum, Integrate vs. NIntegrate, etc.

Solve[a x^2 + b x + c ==0,x]

NSolve[Pi x^5 + E x^3 -x+2.0 ==0,x]

r1=Sum[1/i^2,{i,1,Infinity}]

r2=NSum[1/i^2,{i,1,Infinity}]

(r1-r2)/r1

Integrate[Exp[x]Sin[x],x]

Integrate[Exp[x]Sin[x],{x,0,3}]//Timing

NIntegrate[Exp[x]Sin[x],{x,0,3}]//Timing

NProduct, Solve, NSolve, NDSolve, NMinimize, ...

Recap

f[x_]:= (* general function definition *)

f[x_List]:= (* restricting the type of the arguments *)

f[x_,y_]:= (* multiple arguments *)

f[x_,"a"]:= (* fixing one of the arguments *)

f[x_?NumberQ]:= (* test on argument x *)

f[x_,y_,z_/;condition[z]]:= (* condition on argument z *)

f[x_,y_,z_]/;condition[x,y,z]:=(* condition on arguments *)

f[a_,b__,l___]:=
(* matching one element, a non-empty list of elements, any list  *)

f[a_+b_]:=(* specifying the form of the argument *)

f[{{a_,b_},___}]:=(* again, specifying the structure of the argument *)

f[m:{_^b_}]:=Power[m,-b]:=(* assigning names to parts of a pattern *)

f[a_,b_:Pi]:=Sin[a+b]:= (* default arguments *)

Lists

{a,b,c,d,e,f}

l = {a,b,c,4,Pi,,{},-3.5}

ColumnForm[l]

FullForm[l]

TreeForm[l]

TableForm, MatrixForm

l

l[All]

l[[All]]

l[[1]]

First[l]

l[[0]]

Rest[l]

l[[-1]]

Last[l]

l[[Length[l]]]

Length[l]

l[[8]]
l[[9]]

?Range

?Array

Table[i+j,{i,3},{j,Pi,2Pi,Pi/4}]//MatrixForm

Table[i+j,{i,3},{j,Pi,2Pi,Pi/4}]//TableForm

Table[{i,j,i+j},{i,Pi,2Pi,Pi/4},{j,3}]//Dimensions

M={{1,2,3,4},{5,6,7,8},{9,10,11,12}};

M//MatrixForm

M=Table[((Subscript[A, i])^j),{i,1,5},{j,1,6}]

{{(Subscript[A, 1]),(Subscript[A, 1]^2),(Subscript[A, 1]^3),(Subscript[A, 1]^4),(Subscript[A, 1]^5),(Subscript[A, 1]^6)},{(Subscript[A, 2]),(Subscript[A, 2]^2),(Subscript[A, 2]^3),(Subscript[A, 2]^4),(Subscript[A, 2]^5),(Subscript[A, 2]^6)},{(Subscript[A, 3]),(Subscript[A, 3]^2),(Subscript[A, 3]^3),(Subscript[A, 3]^4),(Subscript[A, 3]^5),(Subscript[A, 3]^6)},{(Subscript[A, 4]),(Subscript[A, 4]^2),(Subscript[A, 4]^3),(Subscript[A, 4]^4),(Subscript[A, 4]^5),(Subscript[A, 4]^6)},{(Subscript[A, 5]),(Subscript[A, 5]^2),(Subscript[A, 5]^3),(Subscript[A, 5]^4),(Subscript[A, 5]^5),(Subscript[A, 5]^6)}}

M=Table[(Subscript[A, i,j]),{i,1,5},{j,1,6}]

MatrixForm[M]

M[[1]]

M[[-2]]

M[[1]][[3]]

M[[1,3]]

Part[M,1,3]

{1,2,34,4}[[1,2]]

(x+y+z)[[2]]

M[[{1,3}]]//MatrixForm

M[[1;;3]]

M[[3,All]]

M[[All,3]]

Dimensions[M]

M[[2;;-2]]

M[[2;;3,2;;3]]

M

Transpose[M]//MatrixForm

Length[M]

Length[M//Transpose]

Dimensions[M]

M={{1,2,3},{4,5,6,10},{7,8,9}}//MatrixForm

{a,b,c}
{{a},{b},{c}}
{{a,b,c}}

Transpose[{{a,b,c}}]

Transpose[{a,b,c}]

l = {a,b,c,4,Pi,,{},-3.5}

Take[l,3]

Take[l,-3]

Take[l,{-3}]

Take[l,{-4,-2}]

Drop[l,-3]

Reverse[l]

Clear[m]

m = {Pi,2,{1,Pi},c,d}

TreeForm[m]

l

m

Join[l,m]

Join[m,l]

Union[l,m]

Length[l]+Length[m]==Length[Union[l,m]]

Intersection[l,m]

Append[l,m]

Prepend[l,m]

m//ColumnForm

ReplacePart[m,{x,y,z},2]//ColumnForm

m

m//ColumnForm

(m = ReplacePart[m,{x,y,z},{3,1}])//ColumnForm

m//ColumnForm

(m1=Insert[m,{1,2,3,4},-1])//ColumnForm

(m2=Insert[m1,{-1,-2,-3,-4},1])//ColumnForm

Delete[m2,{4,2}]//ColumnForm

m2

TreeForm[m2]

Flatten[m2]

Flatten[m2]//TreeForm

Flatten[m2,1]

Flatten[m2,1]//TreeForm

m2

FlattenAt[m2,{4,1}]//TreeForm

FlattenAt[m2,{4,1}]

example[a_,b_,c_]:= a + b^c

partial[example,Pi,2]

out[a_,c_]:=a+Pi^c