Fresnel cosine integral function
fresnelc(
returns
the Fresnel
cosine integral of z
)z
.
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]
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
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))
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 the Fresnel cosine integral function from x = -5
to x = 5
.
syms x fplot(fresnelc(x),[-5 5]) grid on
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
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
Use simplify
to simplify
expressions:
syms x simplify(3*fresnelc(x)+2*fresnelc(-x))
ans = fresnelc(x)
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
.