1. Exponential Integral Table
2. Error Function

Microsoft Excel Exponential Integral Function Approximation. Posted: adminOn. Download Update. Download the. Double click. May 18, 2005  Microsoft Office Application Help - Excel Help. Is it possible to calculate integrals with Excel. Some functions containing integrals using > Excel. Woodwork City pizza of Sarasota Exponential Integral Function Excel. Exponential integral function approximation. Exponential Regression using Microsoft Excel.

In inputbox input x and n
exponential function : e^x=1+ x/1! + x^2/2! +...x^n/n! ; n!=1*2*3..*n
inside the function we can use t(n)=x^n/n! and for recurrence formula t(n)=t(n-1)*x/n
The partial sum to the n-th clause is expressed with s(n) which is ; s(n)=s(n-1)+t(n)
then e=1+s(n)
the calculation will repeat until i=n and t(n) <0.001
the problem is when i run the program the answer is no same with the exponential function above.
Sub e2()
Dim n As Double
Dim x As Double
Dim i As Double
Dim tn As Double
Dim sn As Double
Dim t1 As Double
Dim e As Double
x = CInt(InputBox('x?'))
n = CInt(InputBox('n?'))
Do
For i = 1 To n
t1 = x / (n - 1)
tn = (x / (n - 1)) * (x / n)
sn = tn + t1
e = 1 + sn
If x = 0 Then Exit For
Next i
Loop Until tn < 0.001 Or -tn < 0.001
MsgBox 'e^' & CStr(x) & '=' & CStr(e)
End Sub
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In mathematics, the exponential integral Ei is a special function on the complex plane.It is defined as one particular definite integral of the ratio between an exponential function and its argument.

• 2Properties

Definitions[edit]

For real non zero values of x, the exponential integral Ei(x) is defined as

${\displaystyle \mathrm{Ei}(x)=-{\int }_{-x}^{\mathrm{\infty }}\frac{e^{-t}}{t}dt.}$

The Risch algorithm shows that Ei is not an elementary function. The definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero.

For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and ${\displaystyle \mathrm{\infty }}$.^[1] Instead of Ei, the following notation is used,^[2]

(note that for positive values of x, we have ${\displaystyle -E_1(x)=\mathrm{Ei}(-x)}$). Fated to love you kdrama.

In general, a branch cut is taken on the negative real axis and E₁ can be defined by analytic continuation elsewhere on the complex plane.

For positive values of the real part of ${\displaystyle z}$, this can be written^[3]

${\displaystyle E_1(z)={\int }_1^{\mathrm{\infty }}\frac{e^{-tz}}{t}dt={\int }_0^1\frac{e^{-z/u}}{u}du,\mathrm{\Re }(z)\ge 0.}$

The behaviour of E₁ near the branch cut can be seen by the following relation:^[4]

${\displaystyle \underset{\delta \to 0+}{lim}{E}_1(-x\pm i\delta )=-\mathrm{Ei}(x)\mp i\pi ,x>0.}$

Properties[edit]

Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.

Convergent series[edit]

For real or complex arguments off the negative real axis, ${\displaystyle E_1(z)}$ can be expressed as^[5]

where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant. The sum converges for all complex ${\displaystyle z}$, and we take the usual value of the complex logarithm having a branch cut along the negative real axis.

This formula can be used to compute ${\displaystyle E_1(x)}$ with floating point operations for real ${\displaystyle x}$ between 0 and 2.5. For ${\displaystyle x>2.5}$, the result is inaccurate due to cancellation.

A faster converging series was found by Ramanujan:

${\displaystyle \mathrm{E}\mathrm{i}(x)=\gamma +\ln x+\exp (x/2)\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n-1}{x}^n}{n!2^{n-1}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }\frac{1}{2k+1}}$

These alternating series can also be used to give good asymptotic bounds, e.g.:

${\displaystyle 1-\frac{3x}{4}\le \mathrm{E}\mathrm{i}(x)-\gamma -\ln x\le 1-\frac{3x}{4}+\frac{11x^2}{36}}$

for ${\displaystyle x\ge 0}$.

Asymptotic (divergent) series[edit]

Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, for x = 10 more than 40 terms are required to get an answer correct to three significant figures for ${\displaystyle E_1(z)}$.^[6] However, there is a divergent series approximation that can be obtained by integrating ${\displaystyle z{e}^z{E}_1(z)}$ by parts:^[7]

${\displaystyle E_1(z)=\frac{\exp (-z)}{z}\sum _{n=0}^{N-1}\frac{n!}{(-z{)}^n}}$

which has error of order ${\displaystyle O(N!z^{-N})}$ and is valid for large values of ${\displaystyle \mathrm{Re}(z)}$. The relative error of the approximation above is plotted on the figure to the right for various values of ${\displaystyle N}$, the number of terms in the truncated sum (${\displaystyle N=1}$ in red, ${\displaystyle N=5}$ in pink).

Exponential and logarithmic behavior: bracketing[edit]

From the two series suggested in previous subsections, it follows that ${\displaystyle E_1}$ behaves like a negative exponential for large values of the argument and like a logarithm for small values. For positive real values of the argument, ${\displaystyle E_1}$ can be bracketed by elementary functions as follows:^[8]

Exponential Integral Table

The left-hand side of this inequality is shown in the graph to the left in blue; the central part ${\displaystyle E_1(x)}$ is shown in black and the right-hand side is shown in red.

Definition by Ein[edit]

Both ${\displaystyle \mathrm{Ei}}$ and ${\displaystyle E_1}$ can be written more simply using the entire function${\displaystyle \mathrm{Ein}}$^[9] defined as

${\displaystyle \mathrm{Ein}(z)={\int }_0^z(1-e^{-t})\frac{dt}{t}=\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}{z}^k}{kk!}}$

(note that this is just the alternating series in the above definition of ${\displaystyle {\mathrm{E}}_1}$). Then we have

${\displaystyle \mathrm{Ei}(x)=\gamma +\ln x-\mathrm{Ein}(-x)x>0}$

Relation with other functions[edit]

Mortal kombat annihilation full movie mp4 download in dual audio. Kummer's equation

${\displaystyle z\frac{d^{2}w}{d{z}^2}+(b-z)\frac{dw}{dz}-aw=0}$

is usually solved by the confluent hypergeometric functions${\displaystyle M(a,b,z)}$ and ${\displaystyle U(a,b,z).}$ But when ${\displaystyle a=0}$ and ${\displaystyle b=1,}$ that is,

${\displaystyle z\frac{d^{2}w}{d{z}^2}+(1-z)\frac{dw}{dz}=0}$

we have

${\displaystyle M(0,1,z)=U(0,1,z)=1}$

for all z. A second solution is then given by E₁(−z). In fact,

with the derivative evaluated at ${\displaystyle a=0.}$ Another connexion with the confluent hypergeometric functions is that E₁ is an exponential times the function U(1,1,z):

${\displaystyle E_1(z)=e^{-z}U(1,1,z)}$

The exponential integral is closely related to the logarithmic integral function li(x) by the formula

${\displaystyle \mathrm{li}(e^x)=\mathrm{Ei}(x)}$

for non-zero real values of ${\displaystyle x}$.

The exponential integral may also be generalized to

${\displaystyle E_n(x)={\int }_1^{\mathrm{\infty }}\frac{e^{-xt}}{t^n}dt,}$

which can be written as a special case of the incomplete gamma function:^[10]Hrt 2 tv program live in tenn.

${\displaystyle E_n(x)=x^{n-1}\mathrm{\Gamma }(1-n,x).}$

The generalized form is sometimes called the Misra function^[11]${\displaystyle {\phi }_m(x)}$, defined as

${\displaystyle {\phi }_m(x)=E_{-m}(x).}$

Including a logarithm defines the generalized integro-exponential function^[12]

${\displaystyle E_s^j(z)=\frac{1}{\mathrm{\Gamma }(j+1)}{\int }_1^{\mathrm{\infty }}(\log t{)}^j\frac{e^{-zt}}{t^s}dt.}$

The indefinite integral:

${\displaystyle \mathrm{Ei}(a\cdot b)=\iint e^{ab}dadb}$

is similar in form to the ordinary generating function for ${\displaystyle d(n)}$, the number of divisors of ${\displaystyle n}$:

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}d(n)x^n=\sum _{a=1}^{\mathrm{\infty }}\sum _{b=1}^{\mathrm{\infty }}{x}^{ab}}$

Derivatives[edit]

The derivatives of the generalised functions ${\displaystyle E_n}$ can be calculated by means of the formula ^[13]

Note that the function ${\displaystyle E_0}$ is easy to evaluate (making this recursion useful), since it is just ${\displaystyle e^{-z}/z}$.^[14]

Exponential integral of imaginary argument[edit]

If ${\displaystyle z}$ is imaginary, it has a nonnegative real part, so we can use the formula

${\displaystyle E_1(z)={\int }_1^{\mathrm{\infty }}\frac{e^{-tz}}{t}dt}$

to get a relation with the trigonometric integrals${\displaystyle \mathrm{Si}}$ and ${\displaystyle \mathrm{Ci}}$:

${\displaystyle E_1(ix)=i\left[-\frac{1}{2}\pi +\mathrm{Si}(x)\right]-\mathrm{Ci}(x)(x>0)}$

The real and imaginary parts of ${\displaystyle {\mathrm{E}}_1(ix)}$ are plotted in the figure to the right with black and red curves.

Approximations[edit]

There have been a number of approximations for the exponential integral function. These include:

• The Swamee and Ohija approximation^[15]

${\displaystyle E_1(x)={\left(A^{-7.7}+B\right)}^{-0.13},}$

where

• The Allen and Hastings approximation ^[15][16]

where

• The continued fraction expansion ^[16]

${\displaystyle E_1(x)=\frac{e^{-x}}{x+\frac{1}{1+\frac{1}{x+\frac{2}{1+\frac{2}{x+\frac{3}{\dots }}}}}}.}$

• The approximation of Barry et al.^[17]

${\displaystyle E_1(x)=\frac{e^{-x}}{G+(1-G)e^{-\frac{x}{1-G}}}\ln \left[1+\frac{G}{x}-\frac{1-G}{(h+bx{)}^2}\right],}$

where:

with ${\displaystyle \gamma }$ being the Euler–Mascheroni constant.

Applications[edit]

• Time-dependent heat transfer
• Nonequilibrium groundwater flow in the Theis solution (called a well function)
• Radiative transfer in stellar and planetary atmospheres
• Radial diffusivity equation for transient or unsteady state flow with line sources and sinks
• Solutions to the neutron transport equation in simplified 1-D geometries.^[18]

Error Function

See also[edit]

Notes[edit]

1. ^Abramowitz and Stegun, p. 228
2. ^Abramowitz and Stegun, p. 228, 5.1.1
3. ^Abramowitz and Stegun, p. 228, 5.1.4 with n = 1
4. ^Abramowitz and Stegun, p. 228, 5.1.7
5. ^Abramowitz and Stegun, p. 229, 5.1.11
6. ^Bleistein and Handelsman, p. 2
7. ^Bleistein and Handelsman, p. 3
8. ^Abramowitz and Stegun, p. 229, 5.1.20
9. ^Abramowitz and Stegun, p. 228, see footnote 3.
10. ^Abramowitz and Stegun, p. 230, 5.1.45
11. ^After Misra (1940), p. 178
12. ^Milgram (1985)
13. ^Abramowitz and Stegun, p. 230, 5.1.26
14. ^Abramowitz and Stegun, p. 229, 5.1.24
15. ^ ^abGiao, Pham Huy (2003-05-01). 'Revisit of Well Function Approximation and An Easy Graphical Curve Matching Technique for Theis' Solution'. Ground Water. 41 (3): 387–390. doi:10.1111/j.1745-6584.2003.tb02608.x. ISSN1745-6584.
16. ^ ^abTseng, Peng-Hsiang; Lee, Tien-Chang (1998-02-26). 'Numerical evaluation of exponential integral: Theis well function approximation'. Journal of Hydrology. 205 (1–2): 38–51. Bibcode:1998JHyd.205..38T. doi:10.1016/S0022-1694(97)00134-0.
17. ^Barry, D. A; Parlange, J. -Y; Li, L (2000-01-31). 'Approximation for the exponential integral (Theis well function)'. Journal of Hydrology. 227 (1–4): 287–291. Bibcode:2000JHyd.227.287B. doi:10.1016/S0022-1694(99)00184-5.
18. ^George I. Bell; Samuel Glasstone (1970). Nuclear Reactor Theory. Van Nostrand Reinhold Company.

References[edit]

• Abramowitz, Milton; Irene Stegun (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Abramowitz and Stegun. New York: Dover. ISBN978-0-486-61272-0., Chapter 5.
• Bender, Carl M.; Steven A. Orszag (1978). Advanced mathematical methods for scientists and engineers. McGraw–Hill. ISBN978-0-07-004452-4.
• Bleistein, Norman; Richard A. Handelsman (1986). Asymptotic Expansions of Integrals. Dover. ISBN978-0-486-65082-1.
• Busbridge, Ida W. (1950). 'On the integro-exponential function and the evaluation of some integrals involving it'. Quart. J. Math. (Oxford). 1 (1): 176–184. Bibcode:1950QJMat..1.176B. doi:10.1093/qmath/1.1.176.
• Stankiewicz, A. (1968). 'Tables of the integro-exponential functions'. Acta Astronomica. 18: 289. Bibcode:1968AcA..18.289S.
• Sharma, R. R.; Zohuri, Bahman (1977). 'A general method for an accurate evaluation of exponential integrals E₁(x), x>0'. J. Comput. Phys. 25 (2): 199–204. Bibcode:1977JCoPh.25.199S. doi:10.1016/0021-9991(77)90022-5.
• Kölbig, K. S. (1983). 'On the integral exp(−μt)t^ν−1log^mtdt'. Math. Comput. 41 (163): 171–182. doi:10.1090/S0025-5718-1983-0701632-1.
• Milgram, M. S. (1985). 'The generalized integro-exponential function'. Mathematics of Computation. 44 (170): 443–458. doi:10.1090/S0025-5718-1985-0777276-4. JSTOR2007964. MR0777276.
• Misra, Rama Dhar; Born, M. (1940). 'On the Stability of Crystal Lattices. II'. Mathematical Proceedings of the Cambridge Philosophical Society. 36 (2): 173. Bibcode:1940PCPS..36.173M. doi:10.1017/S030500410001714X.
• Chiccoli, C.; Lorenzutta, S.; Maino, G. (1988). 'On the evaluation of generalized exponential integrals E_ν(x)'. J. Comput. Phys. 78 (2): 278–287. Bibcode:1988JCoPh.78.278C. doi:10.1016/0021-9991(88)90050-2.
• Chiccoli, C.; Lorenzutta, S.; Maino, G. (1990). 'Recent results for generalized exponential integrals'. Computer Math. Applic. 19 (5): 21–29. doi:10.1016/0898-1221(90)90098-5.
• MacLeod, Allan J. (2002). 'The efficient computation of some generalised exponential integrals'. J. Comput. Appl. Math. 148 (2): 363–374. Bibcode:2002JCoAm.138.363M. doi:10.1016/S0377-0427(02)00556-3.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), 'Section 6.3. Exponential Integrals', Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN978-0-521-88068-8
• Temme, N. M. (2010), 'Exponential, Logarithmic, Sine, and Cosine Integrals', in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR2723248

External links[edit]

• Hazewinkel, Michiel, ed. (2001) [1994], 'Integral exponential function', Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN978-1-55608-010-4
• Weisstein, Eric W.'Exponential Integral'. MathWorld.
• Weisstein, Eric W.'En-Function'. MathWorld.
• 'Exponential integral Ei'. Wolfram Functions Site.
• Exponential, Logarithmic, Sine, and Cosine Integrals in DLMF.