Spheroidal Wave Functions Installation Note To make use of the package SpheroidF.m in Mathematica Ver 3.01, user has to create a folder inside the folder C:/.../Wolfram Research/Mathematica/3.0/AddOns/Applications/ a new folder and name it "ProgrammingInMathematica". Then, copy the file spheroidF.m into this folder. In UNIX system, simply copy it into the user's account. To load the package PC, simply key in: <1/2,kind->"prolate"/"oblate"]". ( REMEMBER TO ENCLOSE THE WORD "prolate" or "oblate" IN INVERTED COMMAS. OTHERWISE, WRONG SOLUTION WILL RESULT) . For example, the value of \!\(TraditionalForm\`\(\(S\^\((1)\)\)\_02\)(1.0, 0.5)\) is given by SpheroidSF[0,2,1.0,0.5,Type->1,kind->"prolate"] -0.09875149796 For the corresponding type 2 solution i.e., \!\(TraditionalForm\`\(\(S\^\((2)\)\)\_02\)(1.0, 0.5)\), simply change the type to 2, as shown below: SpheroidSF[0,2,1.0,0.5,Type->2,kind->"prolate"] 18.53186048 For the oblate case , change the kind to "oblate" without keying in the -ic for the parameter c, e.g. \!\(TraditionalForm\`\(\(S\^\((1)\)\)\_02\)(\(-1.0\) i, 0.5)\) SpheroidSF[0,2,1.0,0.5,Type->1,kind->"oblate"] -0.1507602618 In similar fashion, the value of \!\(TraditionalForm\`\(\(S\^\((2)\)\)\_02\)(\(-1.0\) i, 0.5)\) is given by : SpheroidSF[0,2,1.0,0.5,Type->2,kind->"oblate"] 18.30891304 The radial function and derivatives are called in similar style. Some styles is given below: The value of \!\(TraditionalForm\`\(\(R\^\((1)\)\)\_01\)(2.0, 1.077)\) SpheroidRF[0,1,2.0,1.077,Type->1,kind->"prolate"] 0.5311334495 The value of \!\(TraditionalForm\`\(\(R\^\((2)\)\)\_01\)(2.0, 1.077)\) SpheroidRF[0,1,2.0,1.077,Type->2,kind->"prolate"] 0.9215297406 The value of \!\(TraditionalForm\`\(\(R\^\((1)\)\)\_01\)(\(-i\)\ 2.0, i\ 1.077)\) SpheroidRF[0,1,2.0,1.077,Type->1,kind->"oblate"] 0.3656916081 The value of \!\(TraditionalForm\`\(\(R\^\((2)\)\)\_01\)(\(-i\)\ 2.0, i\ 1.077)\) SpheroidRF[0,1,2.0,1.077,Type->2,kind->"oblate"] 0.1199412174 The Derivatives The derivatives of the angular functions is called using "SpheroidSFPrime[m,n,c,\[Eta],Type,kind]. e.g. the values of \!\(TraditionalForm\`\(\(S\^\((1)\)'\)\_01\)(1.2, 0.8)\) is given by : SpheroidSFPrime[0,1,1.2,0.8,Type->1, kind->"prolate"] 0.7425534722 the values of \!\(TraditionalForm\`\(\(S\^\((2)\)'\)\_01\)(1.2, 0.8)\) is given by : SpheroidSFPrime[0,1,1.2,0.8,Type->2, kind->"prolate"] -7.728640026 the values of \!\(TraditionalForm\`\(\(S\^\((1)\)'\)\_01\)(\(-i\)\ 1.2, 0.8)\) is given by : SpheroidSFPrime[0,1,1.2,0.8,Type->1, kind->"oblate"] 1.296898262 the values of \!\(TraditionalForm\`\(\(S\^\((2)\)'\)\_01\)(\(-i\)\ 1.2, 0.8)\) is given by : SpheroidSFPrime[0,1,1.2,0.8,Type->2, kind->"oblate"] 5.664231946 For the radial function derivatives , the value of \!\(TraditionalForm\`\(\(R\^\((1)\)'\)\_11\)(2.0, 1.020)\): SpheroidRFPrime[1,1,2.0,1.020,Type->1, kind->"prolate"] 2.994397411 The value of \!\(TraditionalForm\`\(\(R\^\((2)\)'\)\_11\)(2.0, 1.020)\): SpheroidRFPrime[1,1,2.0,1.020,Type->2, kind->"prolate"] 50.18981734 The value of \!\(TraditionalForm\`\(\(R\^\((1)\)'\)\_11\)(\(-i\)\ 2.0, i\ 1.020)\): SpheroidRFPrime[1,1,2.0,1.020,Type->1, kind->"oblate"] -0.2378755792 The value of \!\(TraditionalForm\`\(\(R\^\((2)\)'\)\_11\)(\(-i\)\ 2.0, i\ 1.020)\): SpheroidRFPrime[1,1,2.0,1.020,Type->2, kind->"oblate"] 0.6128247457 EigenValue The package also includes two functions that can be called to given the characteristic values. While one of the function give, just one eignevalue for given m, n , c , the other givens the first n characteristics values For a single eigenvalue. EigenV[1,1,1.0] 2.195548355 gives the values for \!\(TraditionalForm\`\(\[Lambda]\_11\)(1.0)\). On the other hand, EigenVals[1,1,8] "For n="\[InvisibleSpace]1\[InvisibleSpace]" m="\[InvisibleSpace]1 \[InvisibleSpace]" the Characteristic Value is \ "\[InvisibleSpace]2.195548355 "For n="\[InvisibleSpace]2\[InvisibleSpace]" m="\[InvisibleSpace]1 \[InvisibleSpace]" the Characteristic Value is \ "\[InvisibleSpace]6.424699144 "For n="\[InvisibleSpace]3\[InvisibleSpace]" m="\[InvisibleSpace]1 \[InvisibleSpace]" the Characteristic Value is \ "\[InvisibleSpace]12.46791533 "For n="\[InvisibleSpace]4\[InvisibleSpace]" m="\[InvisibleSpace]1 \[InvisibleSpace]" the Characteristic Value is \ "\[InvisibleSpace]20.48169601 "For n="\[InvisibleSpace]5\[InvisibleSpace]" m="\[InvisibleSpace]1 \[InvisibleSpace]" the Characteristic Value is \ "\[InvisibleSpace]30.48806579 "For n="\[InvisibleSpace]6\[InvisibleSpace]" m="\[InvisibleSpace]1 \[InvisibleSpace]" the Characteristic Value is \ "\[InvisibleSpace]42.49157795 "For n="\[InvisibleSpace]7\[InvisibleSpace]" m="\[InvisibleSpace]1 \[InvisibleSpace]" the Characteristic Value is \ "\[InvisibleSpace]56.49372974 "For n="\[InvisibleSpace]8\[InvisibleSpace]" m="\[InvisibleSpace]1 \[InvisibleSpace]" the Characteristic Value is "\[InvisibleSpace]72.4951466 gives the values for \!\(TraditionalForm\`\(\[Lambda]\_11\)(1.0)\),\!\(TraditionalForm\`\(\[Lambda]\_12\)(1.0)\),\!\(TraditionalForm\`\(\[Lambda]\_13\)(1.0)\),\!\(TraditionalForm\`\(\[Lambda]\_14\)(1.0)\),\!\(TraditionalForm\`\(\[Lambda]\_15\)(1.0)\),\!\(TraditionalForm\`\(\[Lambda]\_16\)(1.0)\),\!\(TraditionalForm\`\(\[Lambda]\_17\)(1.0)\),\!\(TraditionalForm\`\(\[Lambda]\_18\)(1.0)\), i.e the first 8 characteristics values given m, c. The format for the two Functions are 1) EigenV[m,n,c] 2) EigenVals[m,c,n] Reliability The values computed by this package has been verified against all available table in the text "Spheroidal Wave Functions" by Carson Flammer and is found to be accurate and comparable. The tables that were verified is compiled in the file TableI, TableII, and TableIII. Pls kindly refer to the file for proof of accuracy. Thank you. For any error found, please kindly forward to the student author. Any comment is welcomed.