{
    "id": "laplace_transform_cheat_sheet",
    "name": "Laplace Transform",
    "description": "Common Transforms and Identities",

    "metadata": {
        "sourceName": "Paul's Notes",
        "sourceUrl" : "http://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf"
    },

    "aliases": [
        "laplace transforms", "laplace", "laplace rules", 
        "laplace identities", "laplace table", "laplace transform table"
    ],

    "template_type": "reference",

    "section_order": [
        "Common Transforms",
        "Identities"
    ],

    "sections": {
        "Common Transforms" : [
            {
                "key": "L( 1 )", 
                "val": "1 / s"
            },
            {
                "key": "L( t^n )",
                "val": "( n! ) / [ s^( n+1 ) ]"
            },
            {
                "key": "L( e^( at ) )",
                "val": "1 / ( s-a )"
            },
            {
                "key": "L( sin( at ) )",
                "val": "a / [ ( s^2 ) + ( a^2 ) ]"
            },
            {
                "key": "L( cos( at ) )",
                "val": "s / [ ( s^2 ) + ( a^2 ) ]"
            },
            {
                "key": "L( tsin( at ) )",
                "val": "( 2as ) / [ [ ( s^2 ) + ( a^2 ) ]^2 ]"
            },
            {
                "key": "L( tcos( at ) )",
                "val": "[ ( s^2 ) - ( a^2 ) ] / [ [ ( s^2 ) + ( a^2 ) ]^2 ]"
            },
            {
                "key": "L( sinh( at ) )",
                "val": "a / [ ( s^2 ) - ( a^2 ) ]"
            },
            {
                "key": "L( cosh( at ) )",
                "val": "s / [ ( s^2 ) - ( a^2 ) ]"
            },
            {
                "key": "L( [ e^( at ) ]sin( bt ) )",
                "val": "b / [ [ ( s-a )^2 ] + ( b^2 ) ]"
            },
            {
                "key": "L( [ e^( at ) ]cos( bt ) )",
                "val": "( s-a ) / [ [ ( s-a )^2 ] + ( b^2 ) ]"
            },
            {
                "key": "L( [ e^( at ) ]sinh( bt ) )",
                "val": "b / [ [ ( s-a )^2 ] - ( b^2 ) ]"
            },
            {
                "key": "L( [ e^( at ) ]cosh( bt ) )",
                "val": "( s-a ) / [ [ ( s-a )^2 ] - ( b^2 ) ]"
            },
            {
                "key": "L( [ t^n ][ e^( at ) ] )",
                "val": "( n! ) / [ ( s-a )^( n+1 ) ]"
            }
        ],
        "Identities": [
            {
                "key": "F(s)",
                "val": "L{ f(t) }"
            },
            {
                "key": "f(t)",
                "val": "(L^-1){ F(s) }"
            }
        ]
    }
}
