Closed form for $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$?

Question

Let $x>1$ and $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$. Does this integral have a closed form ?

Fist point, the integral converges. Indeed let $u=e^{it^{x}}$ and $v=\frac{-i}{x}t^{1-x}$ we have by integration by parts $$\lim_{T\rightarrow+\infty}\int_0^T=\int_0^1 e^{it^x}dt+\frac{i}{x}e^i+\frac{1-x}{x}\int_1^{+\infty}\frac{e^{it^x}}{t^x}dt$$ One can also prove that $f(x)\sim_{+\infty}1.$

Answer 1

For integral $$f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$$

We first change variable from $t$ to $v=-i t^x$ so that $t=v^{1/x}i^{1/x}$ and $dt=i^{1/x}(1/x)v^{-1+1/x}dv$.

The integral now becomes

$$f(x)=i^{1/x}(1/x)\int_0^{+\infty}e^{-v}v^{-1+1/x}dv......(1)$$

Mathematica 7.0 gives:

$$f(x)=(1/x)\exp\left(\frac{i\pi}{2x}\right)\Gamma(1/x)$$

The definition of the $\Gamma(z)$ function is given by:

$$\Gamma(t)=\int_0^{+\infty}e^{-v}v^{t-1}dv......(2)$$

Comparison of (1) against (2) leads to the same result:

$$f(x)=(1/x)i^{1/x}\Gamma(1/x)=(1/x)\exp\left(\frac{i\pi}{2x}\right)\Gamma(1/x)......(3)$$

Answer 2

@FlybyNight

I just need the space in the answer to show that Gamma function as well as special functions like Bessel functions, are considered as analytic expression. Please do not give me the down vote if you do not agree with me.

Thanks- mike

Here is from Wikipedia:

In mathematics, an analytical expression (or expression in analytical form) is a mathematical expression constructed using well-known operations that lend themselves readily to calculation. As is true for closed-form expressions, the set of well-known functions allowed can vary according to context but always includes the basic arithmetic operations (addition, subtraction, multiplication, and division), exponentiantion to a real exponent (which includes extraction of the nth root), logarithms, and trigonometric functions. However, the class of expressions considered to be analytical expressions tends to be wider than that for closed-form expressions. In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded.