A201578 -id:A201578

Search: a201578 -id:a201578

Displaying 1-3 of 3 results found.

page 1

A201564

Decimal expansion of the least x satisfying x^2 + 2 = csc(x) and 0 < x < Pi.

+10
64

4, 6, 7, 5, 8, 0, 9, 4, 4, 0, 6, 3, 4, 7, 1, 3, 6, 7, 3, 6, 1, 4, 1, 9, 2, 7, 0, 7, 6, 6, 8, 6, 5, 3, 8, 8, 5, 9, 4, 0, 2, 5, 3, 7, 2, 6, 6, 9, 2, 4, 9, 0, 6, 6, 7, 9, 2, 9, 5, 5, 6, 8, 3, 7, 6, 1, 2, 1, 9, 5, 2, 4, 9, 1, 3, 8, 9, 8, 3, 8, 0, 4, 3, 4, 5, 9, 4, 1, 1, 8, 5, 8, 8, 3, 2, 8, 8, 2, 4

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

For many choices of a and c, there are exactly two values of x satisfying a*x^2 + c = csc(x) and 0 < x < Pi. Guide to related sequences, with graphs included in Mathematica programs:

a.... c.... x

1.... 1.... A196825, A201563

1.... 2.... A201564, A201565

1.... 3.... A201566, A201567

1.... 4.... A201568, A201569

1.... 5.... A201570, A201571

1.... 6.... A201572, A201573

1.... 7.... A201574, A201575

1.... 8.... A201576, A201577

1.... 9.... A201579, A201580

1.... 10... A201578, A201581

1.... 0.... A196617, A201582

2.... 0.... A201583, A201584

3.... 0.... A201585, A201586

4.... 0.... A201587, A201588

5.... 0.... A201589, A201590

6.... 0.... A201591, A201653

7.... 0.... A201654, A201655

8.... 0.... A201656, A201657

9.... 0.... A201658, A201659

10... 0.... A201660, A201662

1... -1.... A201661, A201663

2... -1.... A201664, A201665

3... -1.... A201666, A201667

4... -1.... A201668, A201669

5... -1.... A201670, A201671

6... -1.... A201672, A201673

7... -1.... A201674, A201675

8... -1.... A201676, A201677

9... -1.... A201678, A201679

10.. -1.... A201680, A201681

1... -2.... A201682, A201683

1... -3.... A201735, A201736

1... -4.... A201737, A201738

Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0. We call the graph of z=g(u,v) an implicit surface of f.

For an example related to A201564, take f(x,u,v)=u*x^2+v-csc(x) and g(u,v) = a nonzero solution x of f(x,u,v)=0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

Index entries for transcendental numbers.

EXAMPLE

least: 0.4675809440634713673614192707668653885...

greatest: 3.0531517225248702118041550531781137...

MATHEMATICA

(* Program 1: A201564, A201565 *)

a = 1; c = 2;

f[x_] := a*x^2 + c; g[x_] := Csc[x]

Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]

r = x /. FindRoot[f[x] == g[x], {x, .46, .47}, WorkingPrecision -> 110]

RealDigits[r] (* A201564 *)

r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]

RealDigits[r] (* A201565 *)

(* Program 2: implicit surface of u*x^2+v=csc(x) *)

f[{x_, u_, v_}] := u*x^2 + v - Csc[x];

t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, 1}]}, {v, 0, 1}, {u, 2 + v, 10}];

ListPlot3D[Flatten[t, 1]] (* for A201564 *)

PROG

(PARI) a=1; c=2; solve(x=0.4, 0.5, a*x^2 + c - 1/sin(x)) \\ G. C. Greubel, Aug 21 2018

CROSSREFS

Cf. A201397, A201565.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Dec 03 2011

STATUS

approved

A201580

Decimal expansion of greatest x satisfying x^2 + 9 = csc(x) and 0 < x < Pi.

+10
3

3, 0, 8, 7, 6, 0, 9, 6, 0, 2, 7, 8, 3, 6, 0, 6, 1, 3, 3, 0, 0, 1, 1, 9, 0, 4, 9, 8, 8, 4, 6, 7, 0, 1, 5, 0, 7, 7, 1, 5, 8, 0, 1, 9, 5, 4, 6, 6, 8, 5, 6, 6, 5, 3, 7, 9, 3, 4, 2, 4, 0, 9, 4, 1, 4, 2, 9, 0, 1, 8, 0, 5, 2, 0, 5, 2, 6, 7, 1, 8, 3, 9, 8, 1, 8, 5, 7, 2, 8, 3, 5, 2, 7, 8, 2, 3, 7, 6, 2

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A201564 for a guide to related sequences. The Mathematica program includes a graph.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

Index entries for transcendental numbers.

EXAMPLE

least: 0.111187649530336552411321691800657533611...

greatest: 3.087609602783606133001190498846701507...

MATHEMATICA

a = 1; c = 9;

f[x_] := a*x^2 + c; g[x_] := Csc[x]

Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]

r = x /. FindRoot[f[x] == g[x], {x, .1, .2}, WorkingPrecision -> 110]

RealDigits[r] (* A201578 *)

r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]

RealDigits[r] (* A201580 *)

PROG

(PARI) a=1; c=9; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ G. C. Greubel, Aug 21 2018

CROSSREFS

Cf. A201564.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Dec 03 2011

STATUS

approved

A201581

Decimal expansion of greatest x satisfying x^2 + 10 = csc(x) and 0 < x < Pi.

+10
3

3, 0, 9, 0, 4, 2, 1, 2, 7, 0, 1, 5, 2, 1, 5, 1, 4, 5, 3, 6, 5, 1, 4, 9, 7, 4, 4, 3, 8, 9, 9, 9, 2, 0, 5, 3, 4, 3, 8, 7, 8, 8, 2, 1, 3, 8, 3, 1, 5, 6, 3, 5, 0, 1, 4, 0, 8, 5, 5, 5, 5, 1, 8, 9, 9, 6, 6, 3, 6, 3, 1, 5, 9, 8, 0, 6, 4, 7, 6, 1, 2, 8, 4, 0, 6, 0, 6, 1, 1, 0, 6, 8, 9, 9, 4, 4, 5, 3, 4

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A201564 for a guide to related sequences. The Mathematica program includes a graph.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

Index entries for transcendental numbers.

EXAMPLE

least: 0.100066884072919309279805384459381115060...

greatest: 3.090421270152151453651497443899920534...

MATHEMATICA

a = 1; c = 10;

f[x_] := a*x^2 + c; g[x_] := Csc[x]

Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]

r = x /. FindRoot[f[x] == g[x], {x, .1, .2}, WorkingPrecision -> 110]

RealDigits[r] (* A201578 *)

r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]

RealDigits[r] (* A201581 *)

PROG

(PARI) a=1; c=10; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ G. C. Greubel, Aug 21 2018

CROSSREFS

Cf. A201564.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Dec 03 2011

STATUS

approved

page 1

Search completed in 0.005 seconds