Evaluating a complex integral

Hello I'm trying to integrate the following function in MATLAB

but it's returing the wrong answer when I try something like

This is what I have tried so far:

fun = @(t,x,y) exp(1i.*(t.^4+x.*t.^2+y.*t));
P = @(x,y) integral(@(t)fun(t,x,y),-Inf,Inf);
P(1,1)

Any help appreciated and many thanks in advance

[madhan ravi]

what’s the exact answer?

[Torsten]

exp(i*(t^4+x*t^2+y*t)) does not tend to 0 as | t| -> Inf. Thus your integral does not exist (at least in the usual sense).

[Michael Devereux]

According to WolframAlpha the answer is 1.20759 + 0.601534 i

Keep in mind it's a complex exponential so there is a finite solution. This is know as the Pearcey Integral. I am more concerned that I have entered the formula incorrectly than the actual integral itself. Is this the correct way to approach the problem.

Answer 1

[Torsten]

format long
fun = @(t,x,y) exp(-t.^4 + 1i.*y.*t - x.*t.^2 + 1i*pi*0.125);    
P = @(x,y) integral(@(t)fun(t,x*exp(-1i*pi*0.25),y*exp(1i*pi*0.125)),-Inf,Inf); 
P(1,1)

Reference:

https://arxiv.org/pdf/1601.03615.pdf

[Michael Devereux]

Many thanks!

Answer 2

[Abhishek Hullur]

. Evaluate around the rectangle with vertices