naginterfaces.library.specfun.exp_​complex

naginterfaces.library.specfun.exp_complex(z)

exp_complex evaluates the exponential function $e^z$, for complex $z$.

For full information please refer to the NAG Library document for s01ea

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s01eaf.html

Parameters

zcomplex

The argument $z$ of the function.

Returns

ezcomplex

The value of the exponential function.

Raises

NagValueError

(errno $5$)

The imaginary part of argument $z$ is so large that the result has no precision: $z=(⟨value⟩,⟨value⟩)$.

Warns

NagAlgorithmicWarning

(errno $1$)

Argument $z$ causes overflow in real part of result: $z=(⟨value⟩,⟨value⟩)$.

(errno $2$)

Argument $z$ causes overflow in imaginary part of result: $z=(⟨value⟩,⟨value⟩)$.

(errno $3$)

Argument $z$ causes overflow in both real and imaginary parts of result: $z=(⟨value⟩,⟨value⟩)$.

(errno $4$)

The imaginary part of argument $z$ is so large that the result is accurate to less than half precision: $z=(⟨value⟩,⟨value⟩)$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

exp_complex evaluates the exponential function $e^z$, taking care to avoid machine overflow, and giving a warning if the result cannot be computed to more than half precision. The function is evaluated as $e^z=e^x(\cos \left(y\right)+i\sin \left(y\right))$, where $x$ and $y$ are the real and imaginary parts respectively of $z$.

Since $\cos \left(y\right)$ and $\sin \left(y\right)$ are less than or equal to $1$ in magnitude, it is possible that $e^x$ may overflow although $e^x\cos \left(y\right)$ or $e^x\sin \left(y\right)$ does not. In this case the alternative formula $sign\left(\cos \left(y\right)\right)e^{x+ln\left(\left|\cos \left(y\right)\right|\right)}$ is used for the real part of the result, and $sign\left(\sin \left(y\right)\right)e^{x+ln\left(\left|\sin \left(y\right)\right|\right)}$ for the imaginary part. If either part of the result still overflows, a warning is returned through argument $\text{errno}$.

If $Im\left(z\right)$ is too large, precision may be lost in the evaluation of $\sin \left(y\right)$ and $\cos \left(y\right)$. Again, a warning is returned through $\text{errno}$.