Special function calculator

Special function calculator

Special functions are also called higher transcendental functions , Or mathematical physical function . Special functions are functions with specific properties , stay Mathematical analysis 、 Physical research 、 It is often used in engineering calculation , It also often appears in mathematical tables . Special function generally refers to the function that the solution of some kind of differential equation cannot be expressed in the finite form of elementary function , Such as Bessel (Bessel) function . Others are functions defined by certain forms of integration , Such as gamma (Gamma) function . There is also the so-called elliptic function from the perspective of the periodicity of the function . The development of computer and its software brings great convenience to the calculation of special functions . Here is a brief list of the following special functions ： Normal distribution 、k Square distribution 、t Distribution 、F Distribution 、 Sine integral 、 Cosine integral 、 Exponential integral 、 The first kind of elliptic integral 、 The second kind of elliptic integral 、 error function 、 Bessel functions of integer order of the first kind 、 The second kind of Bessel functions of integer order 、 Variant of the first kind of Bessel functions of integer order 、 Variant of the second kind of Bessel functions of integer order 、 Gamma function 、 Incomplete gamma function 、 Incomplete Beta function . Finally, the use of a special function calculator is introduced .

(1) Normal distribution

A random variable x The normal distribution function of P(a,,x) Defined as ：

among a It's the mean ,σ Is the standard deviation ,x Is the input variable value .

(2) Chi square distribution

A random variable k Square distribution function （ or ） as follows ：

among x Is the input variable value ,n It's the degree of freedom .

(3)t Distribution function

A random variable t Distribution (Student- Distribution ) The function is defined as follows ：

among n Is the degree of freedom ,t Is a random variable ,t≥ 0,P(0,n)=0,P(∞,n)=1.

(4)F Distribution function

A random variable F- The distribution function is defined as follows ：

(5) Sine integral function

The sine integral function is defined as follows ：

(6) Cosine integral function

The cosine integral function is defined as follows ：

The formula for calculating the cosine integral is ：

(7) Exponential integral

The exponential integral function is defined as follows ：

The formula for calculating the exponential integral is ：

(8) The first kind of elliptic integral

The first kind of elliptic integral is defined as ：

(9) The second kind of elliptic integral

The second kind of elliptic integral function is defined as ：

among ：0≤k≤1.

(10) error function

The error function is defined as ：

(11) Bessel functions of integer order of the first kind

Real variable x Bessel of integer order of the first kind (Bessel)) function Jn(x) Defined as ：

among n Is a nonnegative integer , The integral expression is ：

For the specific calculation method, refer to the relevant literature .

(12) The second kind of Bessel functions of integer order

Real variable x Bessel of the second order of integers (Bessel) function Yn(x) It has the following recurrence relation ：

among x>0,n Is a nonnegative integer . For the specific calculation method, refer to the relevant literature .

(13) Variant of the first kind of Bessel function of integer order

Real variable x The first type of Bessel (Bessel) function In(x) Expressed as ：

among n Is a nonnegative integer ,i Is an imaginary number ,Jn(ix) Is a pure virtual variable (ix) Bessel functions of the first kind . For the specific calculation method, refer to the relevant literature .

(14) Variant of the second kind of Bessel function of integer order

Real variable x The second kind of integer order Bessel (Bessel) function Kn(x) Expressed as ：

among n Is a nonnegative integer ,i Is an imaginary number ,Jn(ix) Is a pure virtual variable (ix) Bessel functions of the first kind ,Yn(ix) Is a pure virtual variable (ix) Bessel function of the second kind .

For the specific calculation method, refer to the relevant literature .

(15) gamma (Gamma) function

Real variable x Gamma (Gamma) The function is defined as ：

For the specific calculation method, refer to the relevant literature .

(16) Incomplete gamma function

The incomplete gamma function is defined as ：

among a>0,x>0. For the specific calculation method, refer to the relevant literature .

(17) Incomplete Beta function

Incomplete Beta (Beta) The function is defined as ：

among a,b>0,0≤x≤1,B(a,b) For beta function , namely ：

For the specific calculation method, refer to the relevant literature .

reference

[1] Xu Shiliang Ed . C Common algorithm assemblies [M]. Beijing tsinghua university press ,1994 year 1 Yue di 1 edition , The first 442-486 page .
[2] Ning Zhi To write . Microsoft C Science and engineering tool library [M]. Beijing Xueyuan press ,1993 year 12 Yue di 1 edition , The first 116-126 page .
[3] [ beautiful ]W.H.Press, S.A.Teukolsky, W.T.Vetterling, B.P.Flannery Writing . Fu Zuyun , Zhao Meina , Ding Yan Equal translation . C Language numerical algorithm program ( The second edition )[M]. Beijing Electronic industry press ,1995 year 10 Yue di 1 edition . The first 174-225 page .
Numerical Recipes in C, The Art of Scientific Computing, Second Edition, Cambridge University Press, 1988, 1992.
[4] 《 Mathematics Handbook 》 Writing group . Mathematics Handbook [M]. Beijing Higher Education Press ,1979 year 5 Yue di 1 edition , The first 587-648 page .

(18) A special function calculator

The following is the interface of a special function calculator , The software runs on Windows operating system , It has the calculation function of the above special functions . From the drop-down box, you can select the type of special function , Enter the corresponding parameters of the special function , Click on “ Calculation ” Button to output the calculation result , Display with red numbers . There are formulas corresponding to special functions in the lower part of the interface , Explain the meaning and limitations of input parameters .

This special function calculator supports 31 Functions , They are ：

1> Normal distribution function
2>t- Distribution function
3>k- Distribution function
4>F- Distribution function
5> error function
6> Sine integral function
7> Cosine integral function
8> Exponential integral function
9> Elliptic integral function of the first kind
10> Elliptic integral function of the second kind
11> Bessel of integer order of the first kind (Bessel) function
12> The second kind of integer order Bessel (Bessel) function
13> Variant of the first kind of integer order Bessel (Bessel) function
14> Variant type II Bessel of integer order (Bessel) function
15> gamma (Gamma) function
16> Incomplete gamma (Gamma) function
17> Incomplete Beta (Beta) function
18> Exponential function
19> Sine function
20> cosine function
21> Tangent function
22> Cotangent function
23> Antisinusoidal function
24> Arccosine function
25> Arctangent function
26> Inverse cotangent function
27> Natural logarithm function
28> With 2 Base logarithmic function
29> With 10 Base logarithmic function
30> power function
31> Factorial n!

The special function calculator runs on MS Windows Under the operating system , Such as Windows XP、9x、2000、Windows10.

Special function calculation examples

Serial number	Function name	Input parameters	The result of the calculation is	contrast MATLAB
1	Normal distribution	a=0, σ=1, x=2.5	0.9937903	normcdf(2.5,0,1)= 0.993790334674224
		a=-1, σ=0.5, x=-10	0.0000000	normcdf(-10,-1,0.5)= 9.740948918937211e-73
		a=-1, σ=0.5, x=0	0.9772499	normcdf(0,-1,0.5)= 0.977249868051821
2	t- Distribution	n=1, t=0.5	0.2951701	2*tcdf(0.5,1)-1= 0.295167235300867
		n=1, t=5	0.8743329	2*tcdf(5,1)-1= 0.874334083621997
		n=2, t=0.5	0.33333	2*tcdf(0.5,2)-1= 0.333333333333333
		n=4, t=5	0.9925096	2*tcdf(5,4)-1= 0.992509566118726
		n=5, t=5	0.9958952	2*tcdf(5,5)-1= 0.995895284019947
3	k Square distribution	n=1, t=0.5	0.5204999	Chi2cdf(0.5,1)= 0.520499877813047
		n=1, t=5	0.9746527	Chi2cdf(5,1)= 0.974652681322532
		n=2, t=0.5	0.2211992	Chi2cdf(5,1)= 0.221199216928595
4	F- Distribution	n1=2, n2=3, f=3.5	0.1643151	1.0-fcdf(3.5,2,3)= 0.164316767251550
		n1=2, n2=3, f=9	0.0539944	1.0-fcdf(9,2,3)= 0.053994924715604
		n1=5, n2=10, f=3.5	0.0434846	1.0-fcdf(3.5,5,10)= 0.043485040659111
		n1=5, n2=10, f=9	0.0018223	1.0-fcdf(9,5,10)= 0.001822281898469
5	error function	x=0	0.0000000	1-erfc(0)=0
		x=0.25	0.2763264	1-erfc(0.25)= 0.276326390168237
		x=2	0.9953223	1-erfc(2)= 0.995322265018953
6	Sine integral	x=0.5	0.4931074
		x=4.5	1.6541404
7	Cosine integral	x=0.5	-0.1777841
		x=10.5	-0.0782840
		x=14.5	0.0655370
8	Exponential integral	x=0.05	-2.4678985
		x=1.25	-0.1464134
		x=-1.25	-0.1464134
9	Elliptic integral function of the first kind	k=0.5, f=0	0.0000000
		k=0.5, f=1.221730	1.2853000
		k=1, f=0.349066	0.3563787
10	Elliptic integral function of the second kind	k=0.5, f=0	0.0000000
		k=0.5, f=0.523599	0.5178822
		k=1, f=1.570796	1.0000000
11	Bessel functions of integer order of the first kind	n=0, x=0.05	0.9993751	besselj(0,0.05)= 0.999375097649469
		n=1, x=50	-0.0975118	besselj(1,50)= -0.097511828125175
		n=5, x=5	0.2611405	besselj(5,5)= 0.261140546120170
12	The second kind of Bessel functions of integer order	n=0, x=0.05	-1.9793110	bessely(0,0.05)= -1.979311000817210
		n=1, x=5	0.1478631	bessely(1,5)= 0.147863143391227
		n=2, x=0.5	-5.4413708	bessely(2,0.5)= -5.441370837174266
13	Variant of the first kind of Bessel function of integer order	n=0, x=0.05	1.0006251	besseli(0,0.05)= 1.000625097663032
		n=1, x=5	24.3356418	besseli(1,5)= 24.335642142450531
14	Variant of the second kind of Bessel function of integer order	n=0, x=0.05	3.1142340	besselk(0,0.05)= 3.114234029471990
		n=1, x=0.5	1.6564411	besselk(1,0.5)= 1.656441120003301
15	Gamma function	x=0.5	1.7724541	gamma(0.5)= 1.772453850905516
		x=2.5	1.3293405	gamma(2.5)= 1.329340388179137
16	Incomplete gamma function	a=0.5, x=0.1	0.3452792	gammainc(0.1,0.5)= 0.345279153981423
		a=0.5, x=1	0.8427008	gammainc(1,0.5)= 0.842700792949715
		a=5, x=1	0.0036598	gammainc(1,5)= 0.003659846827344
17	Incomplete Beta function	a=0.5, b=0.5, x=0	0.0000000	betainc(0,0.5,0.5)=0
		a=0.5, b=5.0, x=0.2	0.8550738	betainc(0.2,0.5,5)= 0.855072394595919
18	Exponential function	x=1	2.7182818
		x=0.5	1.6487213
19	Natural logarithm function	x=2	0.6931472
20	Factorial n!	n=5	120

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