A093580 -id:A093580 - cp's OEIS frontend

A175314 Decimal expansion of exp(Pi) + exp(-Pi).

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

Offset: 2

R. J. Mathar, Apr 01 2010

			23.1839065510430412555035041051202753...

Index entries for transcendental numbers.

Cf. A039661 (exp(Pi)), A093580 (exp(-Pi)), A175315, A334402 (cosh(Pi)).

Maple
```
evalf(exp(Pi)+exp(-Pi)) ;
```

RealDigits[Exp[Pi]+Exp[-Pi],10,120][[1]] (* Harvey P. Dale, Jan 28 2017 *)

exp(Pi) + exp(-Pi) \\ Michel Marcus, Jun 27 2019

2*cosh(Pi) \\ Charles R Greathouse IV, Jan 30 2025

Equals A039661 + A093580.
Equals 2*cosh(Pi).
Equals 10 * Product_{k>=1} (1 + 4/(2*k+1)^2). - Amiram Eldar, Aug 09 2020

A175315 Decimal expansion of exp(Pi) - exp(-Pi).

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Apr 01 2010

			23.0974787145154967559546686307...

N. Kurokawa, M. Wakayama, A generalization of Lerch’s formula, Czechoslovak Math. J. 54 (2004), 941-947.
Index entries for transcendental numbers.

Cf. A039661 (exp(Pi)), A093580 (exp(-Pi)), A175314.

Maple
```
evalf(exp(Pi)-exp(-Pi)) ;
```

RealDigits[Exp[Pi]-Exp[-Pi],10,120][[1]] (* Harvey P. Dale, Aug 28 2019 *)

exp(Pi) + exp(-Pi) \\ Michel Marcus, Jun 29 2019

2*sinh(Pi) \\ Charles R Greathouse IV, Jan 30 2025

Equals A039661 - A093580.

Equals 2*sinh(Pi).

A334401 Decimal expansion of sinh(Pi).

Original entry on oeis.org

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

Offset: 2

Ilya Gutkovskiy, Apr 26 2020

This constant is transcendental.

			(e^Pi - e^(-Pi))/2 = 11.5487393572577483779773343153884...

Index entries for transcendental numbers

Cf. A000796, A001113, A039661, A068470, A073742, A093580, A156648, A323983, A334402.

Mathematica
```
RealDigits[Sinh[Pi], 10, 110] [[1]]
```

Equals Sum_{k>=0} Pi^(2*k+1)/(2*k+1)!.

Equals 2 * Product_{k>=1} (4*k^2+4)/(4*k^2-1).

A334402 Decimal expansion of cosh(Pi).

Original entry on oeis.org

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

Offset: 2

Ilya Gutkovskiy, Apr 26 2020

This constant is transcendental.

			(e^Pi + e^(-Pi))/2 = 11.5919532755215206277517520525601...

Index entries for transcendental numbers

Cf. A000796, A001113, A039661, A068470, A073743, A093580, A175314, A308715, A323982, A334401.

Mathematica
```
RealDigits[Cosh[Pi], 10, 110] [[1]]
```

Equals Sum_{k>=0} Pi^(2*k)/(2*k)!.
Equals Product_{k>=0} (1 + 4/(2*k+1)^2).
Equals Product_{k>=1} (k^2 + 4)/(k^2 + 1). - Amiram Eldar, Aug 09 2020

A135544 Decimal expansion of (-1)^(I Pi).

Original entry on oeis.org

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

Offset: 0

Marvin Ray Burns, Feb 22 2008, Feb 23 2008

			(-1)^(I*Pi) = exp(-Pi)^(Pi) = 0.000051723186...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Doctor Bombelli, The Math Forum, Exp[ -I * Pi] = Cos[Pi] + i Sin[Pi] = -1.

Cf. A057521, A002388, A093580.

N[(-1)^(I Pi), 1000] FullSimplify[(-1)^(I Pi) == Exp[ -Pi]^Pi == (Exp[ -(1/2)*Pi])^(2*Pi) == Sqrt[Exp[ -Pi]^Pi/(Exp[Pi]^Pi)] == Exp[(-1/2*Pi)]^(Gamma[1/6]*Gamma[5/6]) == 1/(Sqrt[Exp[Pi]^(2*Pi)]) == (Exp[ -(1/2)*Pi])^(2*Pi) == Exp[ -Pi^2]]
Join[{0, 0, 0, 0}, RealDigits[(Exp[-Pi])^(Pi), 10, 96][[1]]] (* G. C. Greubel, Oct 18 2016 *)

exp(-Pi^2) \\ Charles R Greathouse IV, Jan 23 2025

real((-1)^(I*Pi)) \\ Charles R Greathouse IV, Jan 23 2025

a(n) = (-1)^(I Pi) = exp(-Pi)^Pi = (exp( -(1/2)*Pi))^(2*Pi) = sqrt(exp( -Pi)^Pi/(exp(Pi)^Pi)) = exp[(-1/2*Pi)]^(Gamma[1/6]*Gamma[5/6]) = 1/(sqrt[exp[Pi]^(2*Pi)]) = (exp[ -(1/2)*Pi])^(2*Pi) = exp[ -Pi^2].

Offset corrected R. J. Mathar, Jan 26 2009

A101748 Decimal expansion of an i^i, namely exp(3*Pi/2).

Original entry on oeis.org

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

Offset: 3

Robert G. Wilson v, Nov 19 2004

This number multiplied by A101749 = A093580.

i^i = exp(-Pi/2 +- 2*k*Pi).

			This i^i = 111.31777...

Cf. A049006, A093580, A101749.

Mathematica
```
RealDigits[E^(3Pi/2), 10, 111][[1]]
```

Edited by Don Reble, Nov 08 2005

A101749 Decimal expansion of one of the values of i^i, namely exp(-5*Pi/2).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Nov 19 2004

A101748 multiplied by this number = A093580.

i^i = exp(-Pi/2 +- 2*k*Pi).

			This i^i = 0.00038820320...

Cf. A049006, A093580, A101748.

Mathematica
```
RealDigits[E^(-5Pi/2), 10, 111][[1]]
```

Edited by Don Reble, Nov 08 2005

A196535 Decimal expansion of Sum_{j=0..oo} exp(-Pi(2j+1)^2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 03 2011

			0.04321391826429779829201838202725...

Jolley, Summation of Series, Dover (1961) eq (114) on page 22.
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1 (Overseas Publishers Association, Amsterdam, 1986), p. 729, formula 14.

Michael I. Shamos, A catalog of the real numbers, (2007), p. 78.

Cf. A093580, A068466, A010767.

(root[4](2)-1)*GAMMA(1/4)/2^(11/4)/Pi^(3/4) ; evalf(%) ;

RealDigits[ EllipticTheta[2, 0, Exp[-4*Pi]]/2, 10, 105] // First // Prepend[#, 0]&  (* Jean-François Alcover, Feb 12 2013 *)

Equals (2^(1/4)-1) * Gamma(1/4) / ( 2^(11/4) * Pi^(3/4) ).

Equals theta2(exp(-4*Pi))/2.

12 more digits from Jean-François Alcover, Feb 12 2013

cp's OEIS Frontend

A175314 Decimal expansion of exp(Pi) + exp(-Pi).

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A175315 Decimal expansion of exp(Pi) - exp(-Pi).

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PARI

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A334401 Decimal expansion of sinh(Pi).

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A334402 Decimal expansion of cosh(Pi).

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A135544 Decimal expansion of (-1)^(I Pi).

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PARI

PARI

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A101748 Decimal expansion of an i^i, namely exp(3*Pi/2).

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A101749 Decimal expansion of one of the values of i^i, namely exp(-5*Pi/2).

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A196535 Decimal expansion of Sum_{j=0..oo} exp(-Pi(2j+1)^2).