fresnels

Fresnel sine integral function

Syntax

fresnels(z)

Description

fresnels(z) returns the Fresnel sine integral of z.

Examples

Fresnel Sine Integral Function for Numeric and Symbolic Arguments

Find the Fresnel sine integral function for these numbers. Since these are not symbolic objects, you receive floating-point results.

fresnels([-2 0.001 1.22+0.31i])

ans =
-0.3434 + 0.0000i   0.0000 + 0.0000i   0.7697 + 0.2945i

Find the Fresnel sine integral function symbolically by converting the numbers to symbolic objects:

y = fresnels(sym([-2 0.001 1.22+0.31i]))

y =
[ -fresnels(2), fresnels(1/1000), fresnels(61/50 + 31i/100)]

Use vpa to approximate the results:

vpa(y)

ans =
[ -0.34341567836369824219530081595807, 0.00000000052359877559820659249174920261227,...
 0.76969209233306959998384249252902 + 0.29449530344285433030167256417637i]

Fresnel Sine Integral for Special Values

Find the Fresnel sine integral function for special values:

fresnels([0 Inf -Inf i*Inf -i*Inf])

ans =
0.0000 + 0.0000i   0.5000 + 0.0000i  -0.5000 + 0.0000i   0.0000 - 0.5000i...
   0.0000 + 0.5000i

Fresnel Sine Integral for Symbolic Functions

Find the Fresnel sine integral for the function exp(x) + 2*x:

syms x
f = symfun(exp(x)+2*x,x);
fresnels(f)

ans(x) =
fresnels(2*x + exp(x))

Fresnel Sine Integral for Symbolic Vectors and Arrays

Find the Fresnel sine integral for elements of vector V and matrix M:

syms x
V = [sin(x) 2i -7];
M = [0 2; i exp(x)];
fresnels(V)
fresnels(M)

ans =
[ fresnels(sin(x)), fresnels(2i), -fresnels(7)]
ans =
[           0,      fresnels(2)]
[ fresnels(1i), fresnels(exp(x))]

Plot Fresnel Sine Integral Function

Plot the Fresnel sine integral function from x = -5 to x = 5.

syms x
fplot(fresnels(x),[-5 5])
grid on

Differentiate and Find Limits of Fresnel Sine Integral

The functions diff and limit handle expressions containing fresnels.

Find the fourth derivative of the Fresnel sine integral function:

syms x
diff(fresnels(x),x,4)

ans =
- 3*x*pi^2*sin((pi*x^2)/2) - x^3*pi^3*cos((pi*x^2)/2)

Find the limit of the Fresnel sine integral function as x tends to infinity:

syms x
limit(fresnels(x),Inf)

ans =
1/2

Taylor Series Expansion of Fresnel Sine Integral

Use taylor to expand the Fresnel sine integral in terms of the Taylor series:

syms x
taylor(fresnels(x))

ans =
(pi*x^3)/6

Simplify Expressions Containing fresnels

Use simplify to simplify expressions:

syms x
simplify(3*fresnels(x)+2*fresnels(-x))

ans =
fresnels(x)

Input Arguments

z — Upper limit on the Fresnel sine integral

numeric value | vector | matrix | multidimensional array | symbolic variable | symbolic expression | symbolic vector | symbolic matrix | symbolic function

Upper limit on the Fresnel sine integral, specified as a numeric value, vector, matrix, or a multidimensional array or as a symbolic variable, expression, vector, matrix, or function.

More About

Fresnel Sine Integral

The Fresnel sine integral of z is

$$\mathrm{fresnels}(z)={\displaystyle {\int }_0^z\sin \left(\frac{\pi t^2}{2}\right)dt}$$

.

Algorithms

The fresnels(z) function is analytic throughout the complex plane. It satisfies fresnels(-z) = -fresnels(z), conj(fresnels(z)) = fresnels(conj(z)), and fresnels(i*z) = -i*fresnels(z) for all complex values of z.

fresnels(z) returns special values for z = 0, z = ±∞, and z = ±i∞ which are 0, ±5, and ∓0.5i. fresnels(z) returns symbolic function calls for all other symbolic values of z.

Version History

Introduced in R2014a