Mellin-Transform Method for Integral Evaluation: Introduction and Applications to Electromagnetics
Mellin-Transform Method for Integral Evaluation: Introduction and Applications to Electromagnetics
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积分评估的 Mellin 变换方法：电磁学简介及应用
DOI：
10.2200/s00076ed1v01y200612cem013
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发表时间：
2007
影响因子：
1.9
通讯作者：
G. Fikioris
中科院分区：
文献类型：
--
作者：
G. Fikioris
本书介绍了精确计算一维定积分的梅林变换法，并举例说明了该方法在电磁学问题中的应用。一旦掌握了基础知识，人们很快就会意识到这种方法非常强大，通常会产生闭合形式的表达式，很难找到其他方法或从通常的积分表中推导出来。然而，与其他方法不同，目前的方法应用起来非常简单；它通常需要费力的计算，但几乎没有独创性。广义超几何函数和Meijer G-函数与Mellin变换法有着密切的联系，在应用Mellin变换法时经常出现。由于这些函数可以由现代数值例程自动处理，因此它们现在比过去有用得多。首先讨论了梅林变换方法和上述两个函数。然后将该方法应用于三个例子，以获得至少在天线/电磁学文献中被认为是新的结果。在第一个例子中，得到了恒流环形天线辐射功率的广义超几何函数的闭式表达式。第二个例子涉及二维缝隙天线的导纳。在这两个例子中，精确的闭合形式的表达式被应用来改进标准天线教科书中的现有公式。在第三个例子中，导出了一个非常简单的积分表达式，该积分出现在最近未发表的关于无界、双轴各向异性介质的研究中。文中还简要讨论了其他实例。
This book introduces the Mellin-transform method for the exact calculation of one-dimensional definite integrals, and illustrates the application if this method to electromagnetics problems. Once the basics have been mastered, one quickly realizes that the method is extremely powerful, often yielding closed-form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals. Yet, as opposed to other methods, the present method is very straightforward to apply; it usually requires laborious calculations, but little ingenuity. Two functions, the generalized hypergeometric function and the Meijer G-function, are very much related to the Mellin-transform method and arise frequently when the method is applied. Because these functions can be automatically handled by modern numerical routines, they are now much more useful than they were in the past. The Mellin-transform method and the two aforementioned functions are discussed first. Then the method is applied in three examples to obtain results, which, at least in the antenna/electromagnetics literature, are believed to be new. In the first example, a closed-form expression, as a generalized hypergeometric function, is obtained for the power radiated by a constant-current circular-loop antenna. The second example concerns the admittance of a 2-D slot antenna. In both these examples, the exact closed-form expressions are applied to improve upon existing formulas in standard antenna textbooks. In the third example, a very simple expression for an integral arising in recent, unpublished studies of unbounded, biaxially anisotropic media is derived. Additional examples are also briefly discussed.