fresnelc

Fresnel cosine integral function

Syntax

fresnelc(z)

Description

fresnelc(z) returns the Fresnel cosine integral of z.

Examples

Fresnel Cosine Integral Function for Numeric and Symbolic Input Arguments

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

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

ans =
-0.4883 + 0.0000i   0.0010 + 0.0000i   0.8617 - 0.2524i

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

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

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

Use vpa to approximate results:

vpa(y)

ans =
[ -0.48825340607534075450022350335726, 0.00099999999999975325988997279422003,...
 0.86166573430841730950055370401908 - 0.25236540291386150167658349493972i]

Fresnel Cosine Integral Function for Special Values

Find the Fresnel cosine integral function for special values:

fresnelc([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 Cosine Integral for Symbolic Functions

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

syms f(x)
f = exp(x)+2*x;
fresnelc(f)

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

Fresnel Cosine Integral for Symbolic Vectors and Arrays

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

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

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

Plot Fresnel Cosine Integral Function

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

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

Differentiate and Find Limits of Fresnel Cosine Integral

The functions diff and limit handle expressions containing fresnelc.

Find the third derivative of the Fresnel cosine integral function:

syms x
diff(fresnelc(x),x,3)

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

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

syms x
limit(fresnelc(x),Inf)

ans =
1/2

Taylor Series Expansion of Fresnel Cosine Integral

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

syms x
taylor(fresnelc(x))

ans =
x - (x^5*pi^2)/40

Simplify Expressions Containing fresnelc

Use simplify to simplify expressions:

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

ans =
fresnelc(x)

Input Arguments

z — Upper limit on Fresnel cosine integral

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

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

More About

Fresnel Cosine Integral

The Fresnel cosine integral of z is

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

Algorithms

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

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