Mathematical Software. Mathematical Research. Mathematical Education. Tvalx Products.
Complex Number Calculator Precision 45 for Windows 7, Windows Vista, Windows XP, Windows Server 2008, Windows Server (2000, 2003, 2008, 2012), and Windows 2000.
Complex Number Calculator Precision 45 has backward compatibility with College Scientific Calculator series, Scientific Calculator Precision 54, Scientific Calculator Precision 63, Scientific Calculator Precision 72, Scientific Calculator Precision 81, Scientific Calculator Precision 90, Complex Calculator Precision 18, Complex Calculator Precision 27, and Complex Calculator Precision 36. Any formula which works in those calculators will work in this calculator. For that some buttons are duplicated. Button Mod stands for modulus and works on the same way as abs button. Button mod stands for modulo. Button Log stands for principal value of complex Log and is duplicated by button ln. Button log(z) works as Log(z)/ln(10) for complex z and as decimal logarithm for real z, although in Complex Analysis log denotes multi-valued function log(z)=Log(z)+2ni.
This calculator follows classical approach when uncertainty of f(x) calculation is estimated by formula max|(derivative(f))|*|x*uncertainty(x)|, where maximum of function derivative is considered on interval [x-uncertainty(x),|x+uncertainty(x)], and uncertainty(x)=|x|*10^(-precision).
Lets continue. You can type into Edit Formula window a mathematical expression of any length and complexity. For example, type (1+sin(2+cos(3))+tan(4))/(ln(5)-tan(6)+atan(7)). Typing of such expression takes time. If you want to repeat such formula (after other calculations), go to Tab History. In the History rich-text-box find the formula and select it (pressing left button on mouse and dragging mouse). Right-Click and choose Copy from right-click menu. Return to Tab Formula. Right-click into Edit windows and from context-menu choose Paste. All text-boxes in the calculator have similar right-click menus.
Open tab Variables. There are ten variables available. Type into text-boxes any numbers you want to use often in your formulas. Press Parse. Return into tab Formula and type formulas with variables. For example x0+cos(x1)+sin(x2)+tan(x3).
Open the tab Common Constants. There is the list of constants common in science. This list is prebuilt but you can change it and save as a text file. At any moment you can open your list and use it. The list User Constants has similar purpose. Rules for User Constants are weaker. You can copy a part of Common Constants into User Constants. A long list of Common Constants can slow down calculations. If you need only a small part of Common Constants then copy them into User Constants and enable them. Use menu Edit for Cut, Copy, and Paste in Common Constants and User Constants textboxes.
The easiest way to edit formula is left-clicking buttons. It allows to keep brackets balanced, functions names correct and so on. Clicking the button "calculate" triggers calculation of entered formula. The result of calculation appears in the window (text-box) named Result.
The second way is to use keyboard (and keypad). All controls usual for editing are available. Pressing the key Enter triggers calculation. Before using keyboard don't forget to click inside text-box to get focus (blinking cursor).
After calculation the entered formula is not deleted from the Edit window allowing to modify formula. If you want to delete formula select it by mouse and delete. For selecting text you can use right-click menu "Select All" or left-click mouse dragging along the text. For deleting selected text use right-click menu "Cut" or "Delete".
Using right-click menu you can copy and paste text between Edit window and all other text-box windows.
For copying text from saved History file open saved History file (usually in WordPad, Notepad, or MS Word ), drag mouse along the text for selection and then choose Copy from right-click menu. Then go to Formula tab, right-click onto Edit window, select command Paste.
Apply the same procedure for copying text from History window or saved History file into variables windows in Variables tab.
Functions and operations have to be entered exactly as they appears by pressing buttons. Alternative names are not supported.
Numbers can be entered in wide variety of formats. But for exponent always use E, since "e" is reserved for "number e". Long numbers will be rounded for 45 digits. For example, 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 will become 1.23456789012345678901234567890123456789012345678901235E99. Note the last digits 1235. The last 5 appears as result of rounding 12345... . In default Mixed mode (until Scientific mode check-box is checked) integer numbers appears "as is" up to 63 digits. If Scientific check-box is checked then all numbers in variable-boxes and result-box are given in scientific format 1.23456789012345678901234567890123456789012345678901234En, where n has maximum 9 digits, from E-999999999 to E999999999. Numbers with greater exponents will be given status Infinity. Exponents E+9...9 and E9...9 are the same.
In general, the calculator is a complex number calculator and works with complex numbers, but also can be used as a real number calculator, that is a scientific calculator. However, some real numbers, which are infinite sequences of digits, are replaced by finite sequences. Thus the calculator does not distinguish number π, which is infinite sequence of digits, and finite length sequence +3.14159265358979323846264338327950288419716939937510582E0.
The greatest exponent available is E999999999 (nine nines). Numbers with greater exponents are given status Infinity (what is a big simplification, of course). Numbers with negative exponent less than E-999999999 are given status Zero. Number Infinity is extended number with special properties. Number Zero is a usual real number and a special numbers as well. Other special numbers are Uncertainty and NaN. We get Uncertainty dividing Zero by Zero, for example. We get NaN by taking square root of -1, for example. Direct entry of special numbers into edit text-box is not allowed, but you can experiment with special numbers, using 1/0, 0/0, (-1)^0.5, log(-1), and so on.
Arithmetic of special numbers:
0/0=Uncertainty, Infinity/Infinity=Uncertainty, Infinity+Infinity=Uncertainty, Infinity-Infinity=Uncertainty, 0*Infinity=Uncertainty, f(Uncertainty)=Uncertainty, Uncertainty+any=Uncertainty, Uncertainty-any=Uncertainty, Uncertainty*any=Uncertainty, Uncertainty/any=Uncertainty, any/Uncertainty=Uncertainty.
1/0=Infinity, 1/Infinity=0, Infinity*0=Uncertainty, Infinity*Infinity=Infinity, periodic function f(Infinity)=Uncertainty,2^Infinity=Infinity, 1^Infinity=1, (-1)^Infinity=NaN, log(Infinity)=Infinity, log(0)=Infinty.
{Infinity)!=Uncertainty, because (x)! has different behavior for positive and negative x.
2^Infinity=Uncertainty.
Permutations are calculated according to formula P(n;k) = n! / (n - k)! . Note that despite this equality the calculation of P(n;k) is done much faster than calculation of n! / (n - k)! . This is because permutation has a known calculation algorithm, which is built into the program. Whereas the formula n! / (n - k)! calls the factorial procedure two times. Moreover n! grows fast with increase of n and can quickly cause overflow (overflow is a process of losing precision of calculations). Internal algorithm of P(n;k) does not create overflow. The same consideration applies to C(n;k), N(x;k), and G(x;k;q) .
Combinations are calculated according formula C(n;k) = n! / ( k! * (n -k)! ) . They are called also binomial coefficients, because they represents coefficients in polynomial (binomial) (x+y)^n .
The Newton polynomial is given by formula N(x;k) = x(x-1)(x-2)...(x-k+1)/k! . If x is given a real value, it becomes a generalized binomial coefficient. If x is a natural number n, it becomes C(n;k) . For complex k IntegerPart(Modulus(k)) is taken.
G(x;k;q) are generalized Gaussian binomials called also Gaussian coefficients and q-binomial coefficients . The calculation formula is G(x;k;q) = (1-q^x)(1-q^(x-1))...(1-q^(x-k+1))/(1-q)(1-q^2)...(1-q^k) . The x, k, and q can be complex numbers. When the second argument k is complex IntegerPart(Modulus(k)) is taken. For example, G(4+i; 2.3+i; 0.5+i) = (1-(0.5+i)^(4+i))*(1-(0.5+i)^(3+i))/((1-(0.5+i))(1-(0.5+i)^2))
Functions Abs and Mod are identical. They work as modulus(z).
Functions Floor, Ceiling, and factorial works as real function for modulus(z).
Functions Sign works as real function for real part of z, that is sign(z) returns sign of z.Re .
Gamma function is calculated by Spouge algorithm. The algorithm is relatively long and involves many divisions what makes precision relatively low. In order to estimate the precision of calculation use property Gamma(z)=(z-1)! when z is positive integer.
Lower Incomplete Gamma function is calculated by expansion LIGamma(a,z) = Σ( ((-1)^k/k!) * (z^(a+k)/(a+k)) ) = Σ(0; infinity; (-1)^k*z^(a+k) / (k!*(a+k)) ) .
Upper Incomplete Gamma function is calculated by formula UIGamma(a,z) = Gamma(a) - LIGamma(a,z) . Precision of calculation is the same as for Gamma .
Lower Regularized Gamma function is calculated by formula PGamma(a,x) = LIGamma(a,x) / Gamma(a) . Precision of calculation is the same as for Gamma .
Upper Regularized Gamma function is calculated by formula QGamma(a,x) = 1 -PGamma(a,x) . Precision of calculation is the same as for Gamma .
Pi function is calculated by formula Pi(x) = Gamma(x+1) . Precision of calculation is the same as for Gamma .
Sinc function, denoted in the calculator by Sa, is calculated by formula Sa(x) = sinc(x) = sin(x)/x . Sa has removable singularity at zero. So Sa(0)=1 .
Normalized sinc function, denoted in the calculator by NSa, is calculated by formula NSa(x) = sinc(pi*x) = sin(pi*x)/(pi*x) . NSa has removable singularity at zero. So NSa(0)=1 .
Euler-Mascheroni constant γ is represented in Complex Number Calculator Precision 45 by finite length number 5.77215664901532860606512090082402431042159336E-1. Euler-Mascheroni constant γ is used in calculations of some special functions .
Beta function is calculated by formula Beta(a, b) = Gamma(a) * Gamma(b) / Gamma(a + b) . Precision is the same as for Gamma .
Incomplete Beta function is calculated by formula IBeta(z; a; b) = (z^a / a) * 2F1(a, 1-b ,a+1, z) = (z^a/a) * Σ(0; infinity; (a)(a+1)...(a+n-1)(1-b)(1-b+1)...(1-b+(n-1)) / (a+1)...(a+n)) * z^n/n! where 2F1 is a hypergeometric function . Precision of calculation is about 88 digits .
Regularized Incomplete Beta function is calculated by formula RIBeta(z; a; b) = IBeta(z; a; b) / Beta(a, b) . Precision is the same as for Gamma.
Sine Integral function is calculated by Taylor (Maclaurin) series Si(x) = Σ(0; N; (-1)^n*x^(2n+1)/[(2n+1)*(2n+1)!]) = x - x^3/[3!3] + x^5/[5!5] - x^7/[7!7] - ... for |x| <= 55 and by asymptotic approximation for |x| > 55 . Precision of calculation is about 44 digits for |x| < 10, 36 digits for |x| < 30, 27 digits for |x| < 55, 26 digits for 55 < |x| <60, then precision slowly increases while Si(x) is approaching asymptote π/2 on the right and -π/2 on the left.
Lower Sine Integral function is calculated by formula si(x) = Si(x) - π / 2 . Precision is the same as for Si(x).
Σ has syntax Σ(index start; index end; expression). Index start and end are in general any integer numbers. They may also be any formulas not involving variable k. Then the formulas are evaluated and floor of the result is taken. For example Σ(35/10; 40.4; x0^k/k!) is the same as Σ(3; 40; x0^k/k!) . The expression in Σ(index start; index end; expression) is any formula in general involving variable k, but not involving other Σ or Π . For example Σ(0; 20; P(20; 20-k)*x0^k/k!) . So Σ and Π do not allow nesting.
All three arguments can be complex numbers. But for first and second argument IntegerPart(Modulus) is taken. For example , Σ(1;5;1+ik) = +5+i15 and Σ(1;3+4i;1+ik) = +5+i15, since Modulus(3+4i)=5.
When the difference between index start and index end is big and the expression is long then the calculation can be long. If you want to abort calculation, click button Abort on the Menu bar.
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