SetOptions[NSolve,WorkingPrecision->100]

Functions Used to Study Roots of Partial Sums

This contains the definitions of RootsOfPartial, ListRootsOfPartial and RootsPlot

RootsOfPartial outputs a list of complex roots of the nth partial sum of expr in terms of complex coordinates

RootsOfPartial[expr_,z_,n_]:=Module[{roots,poly},
poly = Normal[Series[expr,{z,0,n}]];
roots = z/. NSolve[poly==0,z]
]

ListRootsOfPartial gives the Roots of the nth partial sum of expr in list form, where 1st coordinate is the real part and the 2nd coordinate is the Imaginary coordinate

ListRootsOfPartial[expr_,z_,n_]:=Module[{roots},
roots = RootsOfPartial[expr,z,n];
roots = Transpose[{Re[roots],Im[roots]}]
]

RootsPlot output a graphic of the roots of the nth partial sum of expr

RootsPlot[expr_,z_,n_]:=Module[{roots,size,list},
 roots = RootsOfPartial[expr,z,n];
size = Max[Abs[roots]];
list = Transpose[{Re[roots],Im[roots]}];
ListPlot[list,
PlotStyle->PointSize[.02],
PlotRange->{{-1-size,1+size},{-1-size,1+size}},
AspectRatio->Automatic]]

Sometimes a function is given just as a Taylor Series Around zeros so the functions PolyRoots, PolyListRoots and PolyPlotRoots are the analogous functions to the above only they read in polynomials

RootsOfPoly[poly_,z_]:=z/.NSolve[poly==0,z]

ListRootsOfPoly[poly_,z_] := Module[{roots},
roots = PolyRoots[poly,z];
roots =Transpose[{Re[roots],Im[roots]}]
]

RootsPlotOfPoly[poly_,z_]:=Module[{roots,size},
roots = PolyRoots[poly,z];
size = Max[Abs[roots]];
roots = Transpose[{Re[roots],Im[roots]}];
ListPlot[roots,
PlotRange->{{-1-size,1+size},{-1-size,1+size}},
AspectRatio->Automatic,
PlotStyle->{PointSize[.02]}
]
]

An Example: Application to the Exponential Function

Here is an example of these functions for the exponential function :

Exp[z]
exppoly10=Normal[Series[Exp[z],{z,0,10}]]

RootsOfPartial[Exp[z],z,10]
ListRootsOfPartial[Exp[z],z,10]

RootsPlot[Exp[z],z,10]
RootsPlotOfPoly[exppoly10,z]

Other

Zeros

In this section we give some pictures of some entire functions without any zeros and some entire functions with zeros.

Entire Functions without any zeros

Exponential

Exp[z]

Exp[polynomila] , Exp[z^n] Exp[z^2+1], Exp[(z-1)(z-2)(z-3)], Exp[z^5+z^4+z^3+z^2+z+1]

exppowertable=Table[Exp[z^k],{k,1,5}]

Table[RootsPlot[exppowertable[[k]],z,50],{k,1,5}]

expsomepolys = {Exp[z^2+1],Exp[(z-1)(z-2)(z-3)],Exp[z^5+z^4+z^3+z^2+1]}

Table[RootsPlot[expsomepolys[[k]],z,50],{k,1,3}]

Entire Functions Not of Finite Order: Exp[Exp[z]], Exp[Sin[z]], Exp[Sin[z]Cos[z]]

notfinitelist = {Exp[Exp[z]],Exp[Sin[z]],Exp[Sin[z] Cos[z]]}

Table[RootsPlot[notfinitelist[[k]], ]]

Some Special Functions

1/Gamma[z]

Sin,Cosine, Hyperbolic Sine, Hyperbolic Cosine

Exponential Sums

1/Gamma[z]

RootsPlot[1/Gamma[z],z,50]

nodes={2 ,  Exp[I Pi/3], -3,  5 Exp[I Pi+ I Pi/4], -1-8I , 5 Exp[- I Pi/4]};
listnodes = Transpose[{Re[nodes],Im[nodes]}];

Graphics[{EdgeForm[Black],LightBlue,Polygon[listnodes]}]

expsum = Sum[Exp[nodes[[k]] z],{k,1,6}]

RootsPlot[expsum,z,200]

Airy Function

AiryAi[z]

RootsPlot[AiryAi[z],z,50]

Erf

Erf[z]

RootsPlot[Erf[z],z,50]

Some functions from Series Representations.

alog[z_,n_] = Sum[z^k /(Log[k+2])^k,{k,0,n}]

polyalog = alog[z,50];

RootsPlotOfPoly[polyalog,z]

aloglog[z_,n_] = Sum[z^k / (Log[Log[k+2]])^k, {k,0,n}]

almostExp[z_,n_]=1+Sum[z^k/( k^k Exp[-k] Sqrt[2 Pi k]),{k,0,n}]

Products of Functions we have already seen