# Question:How to decide the location and type of singularities of a complex function without closed form expression?

## Question:How to decide the location and type of singularities of a complex function without closed form expression?

Maple
Hi all, I am just curious about this problem, which has been in my mind for long time. I have seen some complex-valued functions, which are not in closed-form. For example, the complex-valued functions are defined by ODE: y'(t)=a+b*y(t)+c*(y(t))^2 + ... where the numbers in above ODE are complex-valued and the complex-valued functions are defined by functions of y(t), such as exp(y(t)), etc. Such ODEs can only be solved numerically. Sometimes even a closed-form solution is available, we would rather choose to solve it numerically because the closed-form solution often involves hyper functions and super complicated expressions. But we have to integrate them on the complex plane, following some contour. For numerical stability reasons and for correctness, the contour has to be carefully chosen on certain sides of certain singularities and avoiding branch cuts. For example, in inverse Laplace transform and Bromwich contour integration, it's important to select the contour. So my question is: how do we determine the singularities? For specifically, using an example: y'(t, w)=a+b*y(t, w)+c*(y(t, w))^2, and F(t, w)=exp(y(t, w)), (the partial derivative is with respect to "t"). And here w is complex-valued variable and we want to integrate with respect to "w". Suppose we want to find the singularity point on the lower half complex plane(negative imaginary part) that has the smallest vertical distance to 0. Is there a theoretical method and/or experimental method to find out the such singularity point? Thanks a lot!
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